Modelling Patch Dynamics During Ocean Fertilisation
Transcript of Modelling Patch Dynamics During Ocean Fertilisation
![Page 1: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/1.jpg)
Modelling Patch Dynamics During Ocean Fertilisation
Andrew Crawford Submitted in partial fulfilment of the requirements for the Bachelor of Engineering (Environmental) degree with Honours at the University of Western Australia School of Water Research University of Western Australia November 2003
![Page 2: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/2.jpg)
![Page 3: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/3.jpg)
iii
Abstract In light of global warming, the efficiency of the biological pump has become an increasingly
interesting process in the sequestration of carbon from the atmosphere. Iron enrichment of
high nitrate, low chlorophyll oceanic regions have been shown capable of stimulating growth
and increasing the efficiency of the biological pump in these regions. This study uses a model
incorporating nutrient-limited phytoplankton growth, lateral diffusion and export by sinking
of aggregates to investigate the factors which enable maximum export of carbon from the
mixed layer. Export per unit area and total export increases with increasing maximum growth
rate, fertilising concentration and the length scale of the patch. Growth rate was found to be a
major determinant of the time scales involved in export, including the time at which nutrients
become depleted, and the time to first export. Increasing growth and fertilising concentration
increase maximum export flux, and multiple export peaks may occur when either growth or
fertilising concentration are sufficiently large to allow the critical concentration for export to
be reached before diffusion can dissipate concentrations. Successively fertilising the same
patch demonstrates that export increases as the inter-fertilising period increases. The time
series of export is complex and does not suggest a stable pattern of export in runs conducted
for 18 days. The scale of randomness in the initial phytoplankton distribution has little effect
on the export per unit area generated, however, introducing small scale subpatches within the
fertilising patch significantly increases export by allowing diffusion into areas vacated by
export. Knowing these characteristics in which export is maximized, and the parameters
which characterise the export behaviour, allows for the more efficient implementation and
management of iron enrichment to mitigate global climate change.
![Page 4: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/4.jpg)
iv
Acknowledgements I’d like to thank my supervisors, Anya Waite, Greg Ivey and David Johnson for all their
guidance and support throughout the year.
Anya has aided me all the way through this project, not only in an academic sense but also by
giving me just enough space to go at my own pace, but keeping me on track when I asked for
it. Most importantly though, she has given me the encouragement needed to move forward
whenever my confidence was fading.
David has provided invaluable help in understanding the model and has been great for
bouncing some ideas off. His enthusiasm for the project was fantastic.
Greg has given fluid mechanics know-how and constructive criticism when presented with
‘rough and ready’ material.
Also a host of family and friends for helping me through the year. Mum, Dad, Lisa and Bec;
The Quaternity – Abi, Chris and Emma; All who have ever played soccer with me on Sunday;
and the final year engineering crew who have all suffered together with me.
Also others; some over-enthusiastic third years; Michelle Carey for running a club for which I
was meant to be president; and Luke Brown for quick and cheerful computer support.
![Page 5: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/5.jpg)
v
Table of Contents
Table of Contents v List of Figures vi List of Tables viii Chapter 1 Introduction 1
1.1 Background 1 1.2 Direction of this study 2
Chapter 2 Literature Review 3 2.1 Climate change and the role of the oceans 3 2.2 In-situ iron fertilisation experiments 6
2.2.1 IronEx I 7 2.2.2 IronEx II 9 2.2.3 SOIREE 11
2.3 Modelling an iron-stimulated biological pump 13 2.3.1 Models of phytoplankton dynamics 13 2.3.2 Models of carbon export 14
Chapter 3 Methodology 21 3.1 Model Theory 21 3.2 Application of model theory 22 3.3 Investigation of parameters 23
3.3.1 Basic parameter set 24 3.3.2 Variation of growth and fertilising concentration 24 3.3.3 Variation of spatial distribution 25 3.3.4 Multiple fertilisation events 26
Chapter 4 Results 28 4.1 Variations of growth and fertilising concentration parameters 28
4.1.1 Time scales of export 29 4.1.2 Maximum growth rate 30 4.1.3 Fertilising concentration 31 4.1.4 Variation of the initial phytoplankton distribution length scale of randomness 33 4.1.5 Variation of the fertilising concentration length scale 36 4.1.6 Multiple fertilisation events 40 4.1.7 Multiple fertilisation export time series 41
Chapter 5 Discussion 48 5.1 Interpretation of results 48
5.1.1 Variations of maximum growth rate and fertilising concentration 48 5.1.2 Time scales of export 50 5.1.3 Spatial distributions 51 5.1.4 Multiple fertilisation events 52 5.1.5 Summary of interpretations 54
5.2 Practical implications 55 5.3 Issues concerning iron fertilisation for the mitigation of climate change 59 5.4 Recommendations 60
Chapter 6 Conclusions 63 References 65 Appendix 1 70 Appendix 2 74
![Page 6: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/6.jpg)
vi
List of Figures
Figure 2.1 – Atmospheric and oceanic carbon interactions. (from Chisholm, 2000) 4
Figure 2.2 – Time series of mean export of a nutrient-limited growth-diffusion export model (from Waite and Johnson, 2003) 19
Figure 2.3 – Export simulated from a growth-diffusion-export model and its relationship with patch size and the non-dimensional parameter, Q (from Waite and Johnson, 2003) 19
Figure 3.1 – Example of initial phytoplankton distributions at the four different random length scales input. 26
Figure 3.2 – Nutrient distributions of the three fertilising patch variations used. 26
Figure 4.1 – The effect of variations of maximum growth rate, µmax and fertilising concentration for the three different patch lengths shown. The plots represent export per unit area in the left column, time to nutrient depletion, tN in the central column and tN x µmax in the right column. Derived from 10-run ensemble. 28
Figure 4.2 – Export per unit area per unit concentration of nutrient applied, generated by variations of maximum growth rate and fertilising concentration for three different patch lengths – 10k, 25 km and 50km. 29
Figure 4.3 –Time series of export showing variations with maximum growth rate. 30
Figure 4.4 – The variation of parameters characterising export time series behaviour with maximum growth rate. 31
Figure 4.5 – Time series of export showing variations with fertilising concentration. 32
Figure 4.6 - The variation of parameters characterising export time series behaviour with fertilising concentration. 33.
Figure 4.7 – Export time series for four length scales of randomness in initial phytoplankton distribution. Patch length 25km. 34
Figure 4.8 – Export time series for four length scales of randomness in initial phytoplankton distribution. Patch length 40km. 34
Figure 4.9 - Export time series for four length scales of randomness in initial phytoplankton distribution. Patch length 70km. 34
Figure 4.10 - Export time series for four length scales of randomness in initial phytoplankton distribution. Patch length 85km. 34
Figure 4.11 – Export per unit area generated over 10 days from a single fertilisation for different length scales of randomness of the initial phytoplankton distribution. 35
Figure 4.12 - Export time series for three length scales of fertilising subpatch distribution. Patch length 25km. 36
![Page 7: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/7.jpg)
vii
Figure 4.13 - Export time series for three length scales of fertilising subpatch distribution. Patch length 40km. 36
Figure 4.14 - Export time series for three length scales of fertilising subpatch distribution. Patch length 70km. 37
Figure 4.15 - Export time series for three length scales of fertilising subpatch distribution. Patch length 85km. 37
Figure 4.16 – Export per unit area generated over 10 days from a single fertilisation for different length scales of fertilising subpatch. 37
Figure 4.17 - Maximum export flux occurring in different spatial scales of initial phytoplankton distribution and fertilising subpatch scale and how these vary with the length scale, L. 38
Figure 4.18 - Mean time of export weighted by export flux occurring in different spatial scales of initial phytoplankton distribution and fertilising subpatch scale and how these vary with the length scale, L. 38
Figure 4.19 - Timing of the first export occurring in different spatial scales of initial phytoplankton distribution and fertilising subpatch scale and how these vary with the length scale, L. 39
Figure 4.20 - Timing of the first export occurring in different spatial scales of initial phytoplankton distribution and fertilising subpatch scale and how these vary with the length scale, L. 39
Figure 4.21 – Export per unit area generated by three fertilising distributions – at length scales of L/10 and L/5 – and uniform. Q varied by patch length, L. 41
Figure 4.22 – Export per unit area generated by the uniform fertilising distribution. Q varied by µmax. Note that the range of Q, especially where the downturn occurs, is not shown in Figure 4.21. 41
Figure 4.23 - Ten-run ensemble of export time series for multiple fertilisations at different patch length scales corresponding to Q = 0.0431, 0.0271, 0.0207, 0.0171, 0.0147, 0.0130, 0.0118, 0.0108 respectively. Growth rate is 0.4 /day. The period between fertilisations is 1 day (upper left), 3 days (above) and 9 days (left) and is indicated by the dotted line. 42
Figure 4.24 - Ten-run ensemble of export time series for multiple fertilisations at different patch maximum growth rates corresponding to Q = 0.0936, 0.0624, 0.0468, 0.0374, 0.0312, 0.0267, 0.0234, 0.0208, 0.0187, 0.0170 and 0.0156 respectively. Patch length is 25km. The period between fertilisations is 1 day (upper left), 3 days (above) and 9 days (left) and is indicated by the dotted line. 43
Figure 4.25 - Power spectrum density of export time series of 180 day pass. Analysis of final 160 days. The 6 fertilising periods are shown. 44
Figure 4.26 - Power spectrum density of export time series of 180 day pass. Analysis of final 160 days. 8 patch lengths for a fertilising period of 6 days. 45
Figure 4.27 - Power spectrum density of export time series of 18 day pass of different spatial distributions. 44
![Page 8: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/8.jpg)
viii
List of Tables
Table 3.1 - Inputs into MATLAB code, “diffgrow2.m” 31
Table 3.2 - Outputs from MATLAB model, “diffgrow2.m” 31
Table 4.1 – Number of export peaks in 10-run ensemble. 35
Table 4.2 - Distinguishable periods of export from spectral analysis of export time series. 45
![Page 9: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/9.jpg)
Modelling Ocean Fertilisation Patch Dynamics Introduction
1
Chapter 1 Introduction
1.1 Background The biological pump is the process by which atmospheric carbon, in the form of carbon
dioxide (CO2), is fixed by phytoplankton during photosynthesis and then conveyed to the
deep ocean when they die and sink. In as much as 25% of the world’s oceans the efficiency of
the biological pump is greatly reduced and these areas are known as high nutrient, low
chlorophyll (HNLC) regions (de Baar et al., 1999).
In the past decade several mesoscale in-situ iron enrichment experiments have demonstrated
that the HNLC condition and consequently the inefficiency of the biological pump is due to
the limitation of iron in these areas (Boyd, 2002; Coale et al., 1996; Coale et al., 1998). By
applying iron it is possible to create a phytoplankton bloom and, in at least one of the
experiments it has been shown that this leads to a significant increase in the carbon that is
moved from the atmosphere and ‘exported’ from the mixed layer to the deep ocean. These
experiments have been conducted over scales of 8 to 12 km2 and numerical models have
shown that scales of at least 10km are necessary to generate this carbon export (Waite and
Johnson, 2003).
Iron fertilisation of HNLC iron-depleted regions is gaining interest both for its scientific value
and its potential for mitigating climate change. Atmospheric carbon is believed to be
responsible for the majority of the climate change observed over the previous 1000 years
(Crowley, 2000) and because ocean fertilisation promises to increase the rate that atmospheric
carbon is moved and sequestered in the deep ocean, it is thought that it can be used in the
mitigation of climate.
Whether for scientific research or for mitigating climate change, in-situ iron enrichment
experiments, by necessity, require very large areas, and expensive and time consuming
methodologies. Numerical modelling of the characteristics of phytoplankton dynamics, and in
particular export of carbon to deep waters exist as an attractive option in determining how
![Page 10: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/10.jpg)
Modelling Ocean Fertilisation Patch Dynamics Introduction
2
most efficiently to apply iron in specific natural conditions, and how the resulting patch might
be suspected to behave.
1.2 Direction of this study Waite and Johnson (2003) have developed a simple model that considers three important
processes in phytoplankton dynamics under a nutrient-limited regime – nutrient-limited
growth, lateral diffusion and export by aggregation. They have also highlighted the spatial
and temporal complexity that exists in phytoplankton dynamics, even when simplifying
assumptions are used, and how this complexity leads to carbon export behaviour that is non-
trivial.
This study aims to investigate the parameters that characterise the carbon export behaviour of
the Waite and Johnson (2003) growth-diffusion-export model. In particular, it seeks to
determine the conditions and methodology for applying the fertilising nutrient that most
efficiently generates carbon export from the mixed layer.
![Page 11: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/11.jpg)
Modelling Ocean Fertilisation Patch Dynamics Literature Review
3
Chapter 2 Literature Review
2.1 Climate change and the role of the oceans Since the industrial revolution, anthropogenic activity has unbalanced the process known as
the global carbon cycle, by which carbon is moved and partitioned over the earth. Changing
land use and the mining and consumption of fossil fuels have converted terrestrial carbon into
atmospheric carbon, in the form of the greenhouse gas, carbon dioxide (CO2), at a rate that
has not been seen in the previous 1000 years (Crowley, 2000). The CO2 present in the
atmosphere has increased by more than 25% since 1957 (Wigley and Schimel, 2000) and it
has been determined that CO2 contributes most significantly to the changing global climate
(Crowley, 2000). The effect of this increasing pool of carbon in the atmosphere is an increase
in global temperatures as famously reported from the Mauna Loa research station in 1985
(Bacastow et al., 1985).
Although the atmospheric carbon pool is increasing, it is by no means the largest carbon pool
in the global carbon cycle. The intermediate and deeper layers of the world’s oceans contain
98% (38,100 Gt) of the earth’s carbon (Wigley and Schimel, 2000). Consequently, it is the
oceans that offer the greatest capacity for buffering the atmospheric carbon change and global
temperature increase.
The atmospheric and oceanic carbon pools interact via several pathways, including chemical
gradient processes, biological processes and physical processes such as subduction (Chisholm
1995). Sarmiento and Bender (1994) estimate, however, that 75% of the difference in
dissolved inorganic carbon concentrations between the surface and deep ocean is due to the
biologically-mediated pathway known as the ‘biological pump’. The biological pump is the
process by which CO2 fixed in photosynthesis is transferred to the deeper layers of the ocean,
resulting in temporary or permanent sequestration (storage) of carbon (Figure 2.1).
Atmospheric CO2 is initially fixed by autotrophs, such as phytoplankton, which then either
become senescent and sink out as aggregates or are consumed by herbivores that produce
sinking faecal pellets (Chisholm, 1995). Removing the biological pump, and allowing carbon
![Page 12: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/12.jpg)
Modelling Ocean Fertilisation Patch Dynamics Literature Review
4
to be cycled in the ocean only by physical and chemical processes, would more than double
the CO2 concentration in the atmosphere through carbon release as the ocean equilibrated
(Chisholm, 1995).
The efficiency of the biological pump for carbon sequestration is not uniform across the
world’s oceans. High nitrate, low chlorophyll (HNLC) regions, namely the sub-arctic north-
east Pacific, the equatorial Pacific and the Southern Ocean (Chisholm et al., 2001), have long
provided a puzzling phenomenon for oceanographers because despite the favourable nutrient
conditions for production in these regions, productivity is extremely low (Gran, 1931). In
these regions productivity is not coupled to macronutrient concentrations, and even where
upwelling occurs, the resulting nutrients are not utilised (Martin, 1990). Considering that
these regions make up 25% of the world’s oceans (de Baar et al., 1999) they have attracted
considerable research interest into how they are sustained. This scientific interest has been
further accentuated in the last 30 years by the notion that one might use oceanic carbon
sequestration to mitigate the effects of global climate (Chisholm et al., 2001).
Figure 2.1 – Atmospheric and oceanic carbon interactions. The biological pump, with the fixation and removal of carbon to deep waters and recycling through grazing is shown on the left. Physical processes, subduction downwards and upwelling upwards, are represented on the right. (from Chisholm, 2000)
![Page 13: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/13.jpg)
Modelling Ocean Fertilisation Patch Dynamics Literature Review
5
Several theories have been put forward to address the HNLC phenomenon since its discovery.
It seemed apparent given the adequate light availability in these regions that nutrient
limitation was occurring, but the nutrient of limitation was debated. As early as the mid
1930s, iron was invoked as the limiting nutrient responsible (Hart, 1934), however only
recently has the pervasive contamination that distorted early trace element measurements been
reduced so that such propositions could be accurately tested (Morel et al., 1991). One of the
reasons for the limitation of iron has been that it is poorly recycled in the oceans and must
therefore come from wind-transported dust from terrestrial sources. This process is known as
Aeolian deposition (Duce and Tindale, 2001). The HNLC regions have poor rates of Aeolian
deposition because they are many hundreds of kilometres from a terrestrial arid source (Duce
and Tindale, 2001). Some offshore upwelling regions, with high macronutrient
concentrations, do not exhibit HNLC conditions and these regions coincide with areas of
relatively high Aeolian deposition via long-range transport (Martin, 1990).
It is generally considered (Boyd, 2002, Chisholm et al., 2001., Coale et al., 1998, Cullen,
1995) that what is now termed Martin’s Iron Hypothesis (Martin, 1990) was a major
development in the case for iron limitation in HNLC areas. Martin’s hypothesis states that
oceanic productivity during glacial periods may have been higher due to increased upwelling
of nutrients and greater Aeolian deposition of iron from atmospheric dust loads 10-20 times
greater than at present. Martin (1990) hypothesised that these conditions maximised the
efficiency of the biological pump, transporting much more atmospheric carbon to the
intermediate and deeper oceans than currently occurs and resulting in the lower CO2 levels
observed in the paleogeological ice core record. The low atmospheric CO2 in turn had already
been suggested as a mechanism for preventing radiative heating and ultimately leading to the
freezing of the planet over the short time scales observed. Martin’s Iron Hypothesis framed
iron limitation in oceanic regions within the context of geological CO2 levels, thereby
providing circumstantial evidence and a testable theory for the role of iron in large-scale
oceanic processes.
Several elaborations of Martin’s hypothesis have emerged since it was first proposed as
successive bottle and in-situ iron enrichment experiments refined the knowledge of iron
limitation in the open ocean (Cullen, 1995). Cullen (1995) notes that much of the early focus
of Martin’s ideas was considered in the context of HNLC regions. The outcome from a formal
workshop on the topic in 1990 formulated what Cullen calls “the HNLC-iron hypothesis: “An
increase in the rate of supply of iron to the surface layer of the ocean will reduce to depletion
![Page 14: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/14.jpg)
Modelling Ocean Fertilisation Patch Dynamics Literature Review
6
the unused macronutrients, nitrate and phosphate.” ” (Cullen, 1995, p1336). It was intended
that this hypothesis would provide a testable theory of iron limitation. When the rate of supply
of iron to the surface layer of the ocean was increased in the first in-situ iron enrichment
experiment, IronEx I, but the concentrations of nitrates and phosphates varied very little, there
was good evidence to reject the HNLC-iron hypothesis. Cullen (1995) suggests however, that
neither the test nor the hypothesis were perfect since the increase in the rate of supply of the
iron is not defined by the hypothesis and that the iron infusion applied in IronEx I was too
ephemeral to be considered an appropriate test of the hypothesis.
When later discussions progressed, those inclined towards the ecological implications of
Martin’s hypothesis, independently arrived at what is called the “ecumenical iron hypothesis”
(Cullen, 1995). The ecumenical iron hypothesis states that when iron is scarce, smaller cells,
with greater surface:volume ratios can grow more rapidly than larger cells. Therefore the
population of small cells is not limited by iron, but by microzooplankton grazers whose rapid
growth rates can keep phytoplankton under control. By contrast, large cells are not able to
achieve high growth rates when iron is scarce. Under enrichment, however, they can
assimilate macronutrients at a rate that is too great for mesozooplankton to respond, and their
size prevents them from being consumed by microzooplankton. Although this hypothesis
views the HNLC condition by the vague notion of a grazer controlled iron-limited system,
ultimately it predicts that during the enrichment of HNLC areas, we would expect larger
phytoplankton to dominate (Cullen, 1995).
The testing of these hypotheses formed the impetus behind more than 4 in-situ iron
fertilisation experiments including IronEx I, IronEx II, SOIREE and EisenEx, conducted so
far.
2.2 In-situ iron fertilisation experiments
Martin and his colleagues realised that in vitro bottle experiments that had been executed to
date were inadequate in testing the ecosystem and carbon export consequences of their iron-
limitation hypotheses, in particular the ecumenical iron hypothesis (Coale et al., 1996).
Martin therefore proposed “perform[ing] realistic large-scale Fe enrichment experiments in
which phytoplankton species composition, elemental ratios of C:N:P and Si, δ13C ratios, etc
… can be determined, as well as [investigating] the effects of grazing and associated fecal
pellet production, sinking rate and oxygen consumption processes” (Martin, 1990, p10).
![Page 15: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/15.jpg)
Modelling Ocean Fertilisation Patch Dynamics Literature Review
7
The greatest hindrances to such an experiment lay in the lateral and vertical exchanges
between enriched and ambient waters due to advection and turbulent diffusion (Frost, 1996).
The logistics of the number and accuracy of measurements were also significant, however
more than four in-situ experiments have now been carried out, demonstrating that the
technology and the methodology for successful testing does exist.
2.2.1 IronEx I
Conducted in 1993 in the Equatorial Pacific, this is considered the first in-situ testing of
Martin’s hypothesis. Coale et al. (1998) implicitly refer to the hypothesis tested during this
experiment as “determin[ing] whether iron enrichment in the presence of the entire
community results in an increase in the net new production” (p921). The site near the
Galapagos Islands was chosen as the most favourable location for an initial iron experiment
for a number of reasons (Martin and Chisholm, 1992). Primarily these were its high light
intensities and warm temperatures (~25ºC) which would enable high phytoplankton growth
rates, and the vast oceanographic and biological data already available for the area. In
particular, National Oceanic and Atmospheric Administration (NOAA) drifters suggested that
surface flow paths with eddies were rarely observed. This was important since the physical
coherence of the patch was deemed to be of particular concern for the experiment (Stanton et
al., 1998). The NOAA data suggested that the problem of spreading and streaking of the patch
by turbulent diffusive processes at eddy or frontal boundaries (Garret, 1983) could be avoided
at this site.
Pre-fertilisation testing demonstrated that the site was typical of the equatorial Pacific HNLC
area (Coale et al., 1998). Concentrations of nitrate and Chlorophyll a were 10.8 µM and 0.24
µgL-1 respectively and there was a strong pycnocline, well mixed surface layer to 30m and
low horizontal gradients. Initial concentrations of dissolved and particulate iron were
measured at depleted levels of 0.07 and 0.22 nM respectively (Gordon et al., 1998).
443kg of iron as pharmaceutical grade Fe(II) sulphate was deployed once to a large square
patch approximately 8km × 8km. The choice of spatial distribution was due to sampling
considerations which ruled out other methods such as ‘point source’ (Martin and Chisholm,
1992) or ‘streak’ (Watson et al., 1991) applications.
![Page 16: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/16.jpg)
Modelling Ocean Fertilisation Patch Dynamics Literature Review
8
The quantity of iron released was intended to achieve an ocean concentration of 4nM. This
was double the concentration necessary to achieve maximal phytoplankton growth in
laboratory bottle experiments but was considered necessary to account for possible iron
removal processes in the experiment. It was assumed that the concentration of iron when
released would be mixed through to the depth of the mixed layer within 24 h (Martin and
Chisholm, 1992). The root mean square distance for horizontal diffusion (square root of 2kxt)
in this period was calculated to be ~415m assuming a horizontal eddy diffusion coefficient of
kx ~ 10 km2d-1 (Coale et al., 1998). Therefore fertilisation tracks were separated by 450m.
These factors combined were intended to achieve the 4nM concentrations throughout the
patch after 1 day.
Iron behaviour was as expected. One day after iron release, maximum values in the patch
were 3.6 nM due to horizontal eddy diffusion and convective overturn (Coale et al., 1998).
The concentration of dissolved iron (DFe) decreased rapidly in the core of the patch over the
first four days of the experiment to 0.25nM Fe.
The biological response to fertilisation was dramatic. Productivity increased 3 to 4-fold in all
size fractions. Primary production increased monotonically from 10-15 mg C L-1 d-1 to 48 mg
C L-1 d-1 over three days and chlorophyll increased nearly 3-fold to 0.65 mg L-1. The chemical
response, however, was not correlated to this biological response. Although the biological
changes were sizeable, the magnitude of macronutrient drawdown was less than expected. In
contrast to bottle enrichment experiments in which there is complete drawdown (Coale et al.,
1998), nitrate drawdown was undetectable (<0.2 µM) and carbon dioxide fugacity was only
reduced by 10 µatm (Coale et al., 1996).
Approximately five days after the infusion of iron to the system, the core of the patch was
subducted to a depth of 30-35m beneath a low salinity front. At this depth it was confined to a
5-10m layer just above the thermocline. Although the patch could no longer be sampled by
the ship’s flow through system, the presence of the patch was still detectable from the SF6
signal, its distinct salinity and low light transmission (Coale et al., 1998). Since the SF6 signal
remained constant it is likely that the unfertilised waters did not penetrate and dilute the
subducted patch core (Coale et al., 1998).
Several theories were put forward to explain the subdued geochemical response observed in
IronEx I. These were that (1) iron was lost form the patch, (2) the subduction of the patch to
![Page 17: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/17.jpg)
Modelling Ocean Fertilisation Patch Dynamics Literature Review
9
lower light levels minimised the photodissolution of iron colloids and decreased rates of
bioavailable iron production, (3) zooplankton quickly cropped the increase in phytoplankton
biomass, and (4) another nutrient, such as zinc or silicate, became limiting thus preventing
further growth (Coale et al., 1996)
Although the experimental results were confounded by a subduction event, the results were
heartening. This initial experiment heralded the start of a new wave of oceanographic research
by demonstrating that that oceanographers were no longer restricted to observation but that
the problems associated with large scale in-situ experiments could be overcome.
2.2.2 IronEx II
The second iron fertilisation experiment which occurred in 1995 near 3.5°S, 104°W in the
Pacific, followed very closely the methodologies of IronEx I but tried to address the
hypotheses put forward for the unexpectedly low geochemical response to iron enrichment. It
was important that this experiment recreated the biological responses of IronEx I without the
confounding subduction event (Coale et al., 1996).
Prior to fertilisation, the concentrations of nitrate were typical of the HNLC region (~ 10µM)
and initial iron concentrations were expectedly low, recorded at 0.05 nM (Cavender-Bares et
al., 1999).
At day 0 (29th May), iron, in the form of FeSO4 and a SF6 tracer were applied over a 72km2
rectangular deployment area. This was done by creating streaks 400m apart, which were noted
to merge within 1 day. Mixed layer depth measured over the infusion period was averaged at
25m (Coale et al., 1996). Consequently, the day 1 concentration of iron was 2nM. Subsequent
infusions in days 3 and 7 maintained the iron concentration in the infused patch at
approximately 1nM (Landry et al., 2000).
The mixed layer of the patch increased to 50m by day 11 due to small mixing events. Periodic
increases in nitrate concentrations suggested these mixing events introduced nutrient-rich
waters from below into the patch. The patch also expanded horizontally with time from the
initial 72 km2 to 120 km2, however it retained cohesion throughout the experiment (Coale et
al., 1996).
![Page 18: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/18.jpg)
Modelling Ocean Fertilisation Patch Dynamics Literature Review
10
Chlorophyll a concentrations demonstrated a rapid and monotonic increase from 0.15-0.2
µgL-1 initially to almost 4 µgL-1 on day 9, two days after the last infusion of iron. Following
this peak, concentrations decreased to 0.30 µgL-1 on day 17 (Coale et al., 1996).
The biogeochemical response was significantly more developed than that observed in IronEx
I. Nitrate drawdown was approximately 5 µM, however this may be conservative considering
that some nitrate was probably mixed through from below during mixing events on days 11
and 14. After two days, nitrate drawdown tracked silicate drawdown suggesting that diatom
growth was responsible for most of the nitrate uptake (Coale et al., 1996).
Carbon dioxide drawdown also paralleled nitrate drawdown. Maximum depletion occurred on
day 9 in conjunction with the maxima of most of the other biological and chemical indicators
of growth. The south equatorial Pacific near the site is recognised as a strong source of CO2 to
the atmosphere (fco2 in seawater, 526 ppm; fco2 in the atmosphere, 360ppm), however, iron
enhanced growth enabled a drawdown of about 90 µatm, which significantly reduced
outgassing of CO2 from these waters. However, it is not believed that drawdown was so
severe as to limit carbon (Coale et al., 1996).
Iron was rapidly taken up or removed following each of the infusions (initial and day 3 and 7
‘top ups’). It was noted that as the biomass increased in the patch due to iron addition, the rate
of iron removal also increased (Coale et al., 1996).
The community response to the iron enrichment was a shift towards larger organisms,
particularly diatoms. Diatom biomass increased over 85 times to dominate over the naturally-
abundant small (<5µm) phytoplankton which only doubled in size. Total phytoplankton
abundance increased dramatically since they were able to grow faster than predators could
consume them. This led to an imbalance in the early phase of the iron-induced bloom. The
modest picoplankton biomass increase demonstrates that these were most controlled by
zooplankton grazing. However, diatoms increased because they were too large to be
consumed by the fast-growing microzooplankton and too fast-growing to be controlled by the
slower-growing mesozooplankton (Landry et al., 2000).
Estimates of carbon new production suggest that between 5 and 12 µM C was exported from
the surface layer (Coale et al., 1996). Since community analysis showed a lack of larger
mesozooplankton grazers, which are commonly responsible for producing rapidly sinking
![Page 19: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/19.jpg)
Modelling Ocean Fertilisation Patch Dynamics Literature Review
11
fecal pellets that transport carbon below the mixed layer, Coale et al. (1996) suggests that
grazing export did not remove the surface carbon. More likely is that export occurred by
vertical mixing and sinking of diatom aggregates. This is supported by the removal of the SF6
tracer to erosion at the base of the mixed layer by exchange with waters moving relative to the
advection of the patch and thus spread horizontally within the mixed layer (Coale et al.,
1996).
2.2.3 SOIREE
After the success of the iron enrichment experiments in the HNLC areas of the Pacific, the
next step was to trial the meso-scale perturbation experiment in the Southern Ocean. The
conditions of the Southern Ocean are significantly different to those of the equatorial Pacific,
in its physics, its geochemistry and its ecology/biology (Boyd, 2002). More importantly, an
experiment in the Southern Ocean is the ultimate test of the Iron Hypothesis because the
Southern Ocean has the greatest potential for carbon drawdown. This potential is due to the
size of the Southern Ocean, its vast quantities of unused nutrients, except for iron, and the
intermediate and deep water formation delivering surface waters (Chisholm, 2000). It is also
where the coherence between paleoclimate iron flux and carbon export has been observed
most strongly (Coale et al., 1996). Regardless of the iron hypothesis implications, its potential
for carbon drawdown also makes the Southern Ocean the most important place to implement
large-scale iron enrichment to further the understanding of the Southern Ocean’s ability to
mitigate climate change (Chisholm et al., 2001).
The Southern Ocean study site was chosen to be representative of a broad region of
circumpolar HNLC waters but have small current shear stresses in order to maximise the
timescales for tracking the fertilised patch. It was also necessary to try and balance a
regionally representative depth of the mixed layer, whilst making sure that the mixed layer
depth was not too deep so that phytoplankton would become dually iron and light co-limited,
and also so that the iron/sulphur hexaflouride (SF6) would be overly diluted (Boyd and Law,
2001). Whether a true balance was made between a regionally representative site, in the
horizontal and vertical sense, and one which fitted the criteria for successful experimental
design is difficult to say. The up- and down-welling behaviour of the Southern Ocean as well
as other physical behaviours are highly seasonal and also spatially variable (Trull et al.,
2001). Trull et al. (2001), also members of the SOIREE science team, contend that “less than
half of the Southern Ocean is likely to exhibit a response similar to that which occurred
during SOIREE (because only waters well south of the Polar Front are silica-rich throughout
![Page 20: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/20.jpg)
Modelling Ocean Fertilisation Patch Dynamics Literature Review
12
the year), and that carbon export by bloom subduction is unlikely” (p2440). This is unless
there are significant changes in the community structure or algal physiology (Trull et al.,
2001). Given this, the site chosen is relatively representative of the summer period in which
the experiment was conducted, but will not represent any response seen in the winter at the
same site (Trull et al., 2001). The average mixed layer depth for the chosen site was 65m.
Pre-fertilisation testing showed that at the site, mixed layer nitrate and phosphate, at ~25 ±
1µM and ~1.5 ± 0.2µM respectively, were relatively high. Dissolved iron levels were low at
~0.08 ± 0.03 nM for polar waters and chlorophyll a concentration was 0.25 mg m-3 (Boyd et
al., 2000). These concentrations are indicative of the Southern Ocean.
The methods for conducting a mesoscale iron enrichment experiment were quite well
established at SOIREE, following both IronEx I and II. As in the case of both the earlier iron
enrichment experiments, SF6 was used as a tracer for iron added as acidified FeSO4.7H2O.
Testing was conducted in a Lagrangian framework. The study site was infused with iron to
3.8nM in a patch ~50km2 at day 0 (9th February 1999). Subsequent infusions occurred on day
3, 5 and 7 to ~2.6nM over areas of 32, 33.8 and 38.5 km2 respectively (Boyd and Law, 2001).
The initial patch had mixed through within two days to 100 km2 and within 13 days to ~200
km2. No physical structural changes were observed within the first four days of the
experiment, however, calm days caused the generation of transient and temporary thermal
strata on days 5/6, 8/9 and 13. (Boyd et al., 2000)
Dissolved iron was measured as a criterion for subsequent iron infusions. After the first
infusion, levels were initially increased similar to those in Iron Ex I and II and iron rich areas
of the Atlantic polar front (Boyd et al., 2000). By day 2, however, levels of iron had
decreased rapidly, presumably due to the effects of the patch spreading and conversion to
particulate iron. Total (unfiltered) iron remained at ~2nM. Later infusions followed when
dissolved iron concentrations approached background levels. After the fourth and final
infusion iron levels decreased, but thereafter remained relatively stable at ~1nM until the end
of the study site occupation (Boyd et al., 2000).
The geochemical response was expectedly large. Nitrate drawdown was 3µM and pCO2
drawdown was 35µatm, which, although less than the equatorial pacific equivalents, are still
sizeable (Boyd, 2002). The biological and physiological changes that were observed in both
IronEx I and II were also observed in SOIREE. Ratios of carbon to chlorophyll a halved by
![Page 21: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/21.jpg)
Modelling Ocean Fertilisation Patch Dynamics Literature Review
13
day 13 to ~45 within the patch compared to just prior to the first infusion. The community
structure also changed. There was a floristic shift over the period of the experiment with
initial chlorophyll a increases attributed to pico-eukaryotes, to autotrophic flagellates between
days 2 and 8, and finally to large diatoms (particularly Fragilariopsis kerguelensis) from day
6. These large diatoms, of between 30-50 µm cell length and growing at 4.4 x 104 cells L-1 by
day 12, also increased the number of cells in their diatom chains twofold to 14 by day 12. It is
believed that this floristic shift is responsible for mediating changes in the concentration of
climate-reactive gases in surface waters (Boyd et al., 2000). The dominance by F.
kerguelensis is also believed to be responsible for the low rates of diatom herbivory as it is
morphologically adapted, with a highly silicified skeleton, to minimise grazing. Diatoms
accounted for 75% of production (Boyd et al., 2000)
Although the SOIREE study supported the first tenant of Martin’s iron hypothesis by
demonstrating a significant biological response to iron addition, it did not support the second
tenant in that there was no evidence of increased particle export (Boyd et al., 2000). It is
likely that the lateral diffusion, which amounted to 25% of the algal growth rate, prevented a
single large export event (Waite and Nodder, 2001). Furthermore, sediment traps suggest that
sinking rates and aggregation characteristics did change over the course of the experiment.
This is consistent with Muggli et al. (1996) who have shown that sinking rates are higher
under iron stress. The phenomenon of aggregate formation was increased during iron
enrichment (Waite and Nodder, 2001).
2.3 Modelling an iron-stimulated biological pump
Modelling the biological pump is important in understanding the role that the oceans play in
the global carbon cycle. Accurate modelling requires consideration of aspects of
phytoplankton growth and distribution, and how the carbon fixed by phytoplankton is
conveyed to the intermediate and deep ocean.
2.3.1 Models of phytoplankton dynamics
The early work of Kierstead and Slobodkin (1953) developed a classical model for oceanic
phytoplankton dynamics by balancing horizontal diffusion with growth of phytoplankton in
the mixed layer. They considered a body of water that was favourable to growth bounded by,
and mixing at the edges with, water that is unsuitable. The unsuitable water could be
![Page 22: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/22.jpg)
Modelling Ocean Fertilisation Patch Dynamics Literature Review
14
characterised by any of several parameters including salinity, temperature or nutrients. In the
case of iron enrichment the suitable waters are no longer limited by iron whereas unsuitable
water is still iron-constrained and therefore has relatively small growth rates. By assuming
that diffusive losses were large, or that the water was otherwise unsuitable for concentrations
of phytoplankton outside a particular area of patch, the Kierstead and Slobodkin model
delivers a critical minimum patch size able to be sustained. Subsequent elaborations of this
model incorporated the effect of grazing by zooplankton (Wroblewski et al., 1975; Platt,
1975), the scale dependence of the diffusion coefficient (Platt and Denman, 1975), nutrient
limitation and light periodicity (Wroblewski and O’Brien, 1976).
In its simplest form, considering only horizontal diffusion with spatially constant diffusion
coefficients and by neglecting advection, phytoplankton dynamics within the patch can be
represented by:
QCSCdx
CdKdtdC
h −+= max2
2
µ
where C is the concentration of phytoplankton
Kh is the horizontal diffusion coefficient
µmax is the maximum growth rate
S is the function that details the growth formulation
and; Q is the rate of collective removal of phytoplankton from predation by herbivores,
extracellular release, and sedimentation.
(Wroblewski and O’Brien, 1976)
2.3.2 Models of carbon export
Below the surface or mixed layer it can be assumed that oceanic waters are quiet and that
there is little return of particles to the surface (O’Brien et al., 2003). In modelling the
biological pump one therefore needs only to consider carbon export from this mixed layer.
Sedimentation of particulate organic carbon (POC) is the primary biologically-mediated
method of carbon export from the mixed layer in marine systems. Due to Stokes’ Law larger
particles sink faster than smaller particles or cells and therefore have a significant effect on
the fate of organic matter (Jackson and Lochmann, 1992). They contribute disproportionately
to the carbon flux into deeper waters. Consequently, carbon export models have focussed on
these larger particles and the processes which form them.
![Page 23: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/23.jpg)
Modelling Ocean Fertilisation Patch Dynamics Literature Review
15
Large particles may exist as large phytoplankton, especially diatoms, or as accumulations of
smaller particles, especially smaller phytoplankton (nanoplankton or picoplankton).
Furthermore, these accumulations may be aggregates formed from the collisions between
particles, or as non-aggregate particles formed through mechanisms including algal division
and diatom chain formation.
Aggregates of organic matter (also known as flocs or “marine snow”) are a highly visible
phenomena observed in the wake of an algal bloom (Jackson, 1990). The collision processes
by which they form have been artificially replicated in the laboratory with naturally occurring
organic matter (Waite et al., 1997). Kinetic coagulation theories developed in understanding
particle dynamics of lakes, conclude that the rates of algal losses to aggregation (coagulation)
can be comparable to losses caused by zooplankton grazing. This is supported by algal losses
in in-situ iron enrichment experiments in oceanic systems (Boyd et al., 2002; Coale et al.,
1996) and the ecumenical iron hypothesis (Cullen, 1995) which both demonstrate that when
iron is abundant sedimentation losses are at least as important as herbivory losses.
In coagulation theory, an aggregate is a particle formed by collision of two smaller particles,
the largest aggregates being the product of repetitive collisions and coalescence of smaller
ones. Up until Jackson (1990) the concepts of coagulation theory had been applied primarily
to freshwater environments to explain mass fluxes. Jackson applied these principles to a
marine system and expanded the focus of the theory to explain the effect of coagulation and
carbon export on phytoplankton dynamics. In the Jackson model a number of different size
classes were considered. Particles were removed from a particular size class either into a
larger size class following a collision and coagulation, or as they were exported below the
mixed layer. The likelihood of incorporation into larger and larger sized particles, the
collision rate, was given by a function of sizes of colliding particles, their concentrations and
environmental and physiological parameters. Three collision mechanisms were used to
describe particle collisions – Brownian motion, shear (either laminar or turbulent) and
differential sedimentation (in which a larger particle falling faster than a smaller one ‘runs
into the back’ of the smaller one).
The Jackson model used the criteria that the fundamental size class was a solitary cell and the
size of each particle is given by the number of cells it contains. The source of particulate
![Page 24: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/24.jpg)
Modelling Ocean Fertilisation Patch Dynamics Literature Review
16
matter for the model was cell growth, increasing the size of aggregates and increasing the
number of separate solitary cells. The ultimate source is removal by sinking.
An important conclusion of Jackson’s model is that algal systems have a two-state nature,
either coagulation is not important or it is dominant. The distinction between these two states
occurs at some critical cell concentration (Ccr) (Jackson and Lochmann, 1992). This provides
a parameter to compare the likelihood of continued growth (<Ccr) or rapid loss of particles to
coagulation and decrease in particle concentration (>Ccr). The value of this Ccr therefore
becomes important when we are considering the aggregation response of a system and the
relationship between growth as a source and export as a sink. Jackson and Lochmann (1992)
further developed this model by incorporating light and nutrient limitations, which
constrained the growth dynamics of the phytoplankton.
Since coagulating particles can occur when growth occurs even at reasonably constant rates,
there is likely to be a cap placed on phytoplankton concentrations in natural systems because
export from the mixed layer is ongoing (Jackson and Lochmann, 1992). The importance of
coagulation in controlling biomass is not likely to be great in areas where shear is low or
grazing is high. In natural systems, this value of Ccr will roughly correlate with the maximum
phytoplankton concentrations that exist (Jackson and Lochmann, 1992).
Results from SOIREE suggest that one of the primary algal losses from the iron-enriched
bloom was due to diffusive losses at the edges of the patch. The entrainment of surrounding
HNLC waters and subsequent dilution of phytoplankton stocks in the labelled patch was
given as a reason for its longevity. Abraham et al. (2000) report losses due to lateral diffusion
to be 0.1d-1, 75% of net growth rate. Boyd et al. (2001) call this a “physical artefact” in as far
as it obscures the biological processes underlying Martin’s hypothesis. Boyd et al. (2002)
investigate these “physical artefacts” in an aggregation model (Jackson and Lochmann, 1992)
modified by imposing a constant specific algal growth rate representative of both the SOIREE
and IronEx II mesoscale experiments. Two cases were modelled – a standard run,
incorporating best guesses of parameters from the respective enrichment experiments; and a
run with a higher net growth rate. The purpose of the higher net growth rate was in order to
mimic a larger scale enrichment experiment (~100km) where it was assumed that dilution of
bloom stocks via horizontal diffusion is negligible. An algal monoculture was assumed
because of its simplicity and the dominance of a single species in both the IronEx II and
SOIREE studies.
![Page 25: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/25.jpg)
Modelling Ocean Fertilisation Patch Dynamics Literature Review
17
By using a higher net growth rate to represent lower diffusive losses the consideration of
diffusive losses by Boyd et al. (2002) is a simple one. Although the model is based upon a
thorough aggregation model (Jackson and Lochmann, 1992) it fails to consider classical
diffusion-growth theory (Kierstead and Slobodkin, 1953). Boyd et al. (2002) and Boyd et al.
(2001) mistakenly refer to lateral losses of phytoplankton bloom stocks as “lateral advective
losses” (eg – Boyd et al., 2002. pp 36-4) and the authors refer to Abraham et al. (2000) to
provide an estimate of this lateral loss. What these authors term “lateral advective losses” are,
however, more accurately referred to as “horizontal diffusive losses”. Indeed, this is the actual
context of Abraham et al. (2000) estimates quoted in Boyd et al. (2002) and Boyd et al.
(2001). In contrast to some of the assertions of Boyd et al. (2001), the concept of diffusive
losses is not uncommon in phytoplankton patch modelling (see Kierstead and Slobodkin,
1953). This paper will refer accordingly to lateral losses as horizontal diffusive losses, in the
more accurate context used by Abraham et al. (2000) and followed by Waite and Johnson
(2003).
The results of Boyd et al. (2002) do accurately reflect the timing of downward particle flux
measurements of both IronEx II and SOIREE. They do not, however, simulate the magnitude
of the changes in particle concentrations and downward flux from observations.
In modelling the role of diffusive losses in mesoscale enrichment experiments, applying a loss
term that is dependent upon length scale provides a more realistic consideration of the process
than merely using a higher net growth rate. This is the approach taken by Waite and Johnson
(2003). A simple horizontal diffusion-growth model is employed for phytoplankton dynamics
(Kierstead and Slobodkin, 1953; Wroblewski and O’Brien, 1976). Horizontal diffusion is
made to be length-scale dependent (Okubo, 1971) and spatially uniform.
A 2-dimensional model is justified by considering only phytoplankton which are uniformly
distributed vertically in the mixed layer. This is justified since sinking rates of unaggregated
phytoplankton are low (O’Brien et al., 2002). Vertical diffusion out of the mixed layer is
assumed to be small also. Vertical export from the mixed layer occurs only through
aggregation and sinking which is modelled as a two-state function (Jackson, 1990)
E= -1Ccr C≥Ccr
0 C<Ccr
![Page 26: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/26.jpg)
Modelling Ocean Fertilisation Patch Dynamics Literature Review
18
This means that once concentration reaches a certain critical threshold there is an
instantaneous export of this amount out of the mixed layer. The rate of sinking of single cells
is considered to be negligible and it is assumed that the timescale of export by aggregation
and sinking is shorter than the timescale of growth (Jackson and Lochmann, 1992). Thus the
phytoplankton dynamics can be modelled according to the following relationship:
ECCKdtdC
h −+∇= µ2
Waite and Johnson (2003) investigate both the non-limited and nutrient-limited cases of
phytoplankton growth by developing the non-dimensional parameter, Q. Q represents the
ratio between diffusion growth and patch length scale.
2LKQµ
=
In both cases total mean export is dependent upon Q. The non-limited case demonstrates
characteristics chaotic patterns in surface patch structure as holes form from sinking in
concentrated areas.
In the nutrient-limited case the growth term, µ, is made to be dependent upon nutrient
concentration via the Monod equation (McCarthy, 1981). Nutrients are modelled in a similar
diffusion model to phytoplankton, diffusing horizontally but not vertically. Nutrient removal
is via uptake by phytoplankton. As phytoplankton sink out it was assumed that they take the
associated nutrients with them so there is no assumed recycling by grazing. Coagulation
theory also suggests that rapid removal of particles moves organic matter to the deep ocean
faster and makes it more likely they will fall rather than be eaten (Jackson and Lochmann,
1992).
In contrast to the non-limited case, nutrient limitation makes µ and therefore Q, time
dependent. Therefore Q is considered to be dependant upon µmax rather than µ in the nutrient-
limited case. Due to the time dependence, export occurs in events (Figure 2.2) with a primary
sedimentation event followed by smaller secondary events as nutrients are removed and
become increasingly limited (Waite and Johnson, 2003).
![Page 27: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/27.jpg)
Modelling Ocean Fertilisation Patch Dynamics Literature Review
19
Figure 2.3 – Export simulated from a growth-diffusion-export model and its relationship with patch size and the non-dimensional parameter, Q. Note that export only occurs after Q ~ 0.7 (from Waite and Johnson, 2003).
Figure 2.2 – Time series of mean export of a nutrient-limited growth-diffusion-export model over 10 passes. At 10 km there is no export generated as the patch is too small. Duration and intensity increases as patch size increases. (from Waite and Johnson, 2003).
![Page 28: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/28.jpg)
Modelling Ocean Fertilisation Patch Dynamics Literature Review
20
The total export increases with decreasing Q and therefore with increasing patch size (for
constant µmax) (Figure 2.3). The critical Q at which export begins to occur is 0.07 however the
maximum export occurs far beyond the assumptions of this model. At low values of Q (large
patch size) vertical diffusion out of the patch becomes comparable to the horizontal diffusion
and cannot be neglected.
The Waite and Johnson model highlights the spatial and temporal complexity of export and
phytoplankton concentration in fertilised patches, and therefore the inadequacies in field
sampling, especially in the in-situ enrichment experiments conducted so far. Furthermore, it
demonstrates that such experiments are not likely to be testing the second tenet of Martin’s
iron hypothesis, that is, export to intermediate and deep waters, because patch size is too
small to initiate export.
The temporal and spatial complexities in phytoplankton dynamics implied by the Waite and
Johnson model are inherently interesting, but especially so in the context of how these
complexities may affect the way that future in-situ experiments and non-scientific projects are
conducted. To date, the use of numerical models to simulate carbon export behaviour of a
phytoplankton system has been progressive but needs to be continued. Diffusion-growth-
export models address the key processes identified by in-situ experiments as determining
phytoplankton dynamics. The next step is too fully interrogate such models as to how these
processes create the phytoplankton and export behaviour observed.
![Page 29: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/29.jpg)
Modelling Ocean Fertilisation Patch Dynamics Methodology
21
Chapter 3 Methodology
3.1 Model Theory A diffusion-growth-export model (Waite and Johnson, 2003) was used to simulate carbon
export from the phytoplankton system. This model is based upon a simple nutrient-limited
diffusion-growth equation with export from the mixed layer governed by a two-state function.
The model solves the diffusion-growth equation given below;
ECCKdtdC
h −+∇= µ2
where Kh is a spatially uniform horizontal diffusion coefficient, µ is growth rate and E is an
export function. This model considers only export by aggregation and sinking because the
time scales of sinking by aggregation are considered to be much smaller than those for
sinking of individual cells, or of other mechanisms of carbon removal, such as grazing
(Jackson and Lochmann, 1992).
The export function represents a simple two-state process. Either export is important when the
concentration is over some critical concentration, Ccr, and phytoplankton aggregate and sink,
or it is not significant (Jackson and Lochmann, 1992).
E= -1Ccr C≥Ccr
0 C<Ccr
The model considers only the domain given by the mixed layer. Vertical diffusion is
dominant over the low sinking rates of unaggregated phytoplankton within the mixed layer.
Therefore, the distribution of phytoplankton is considered to be vertically uniform (O’Brien et
al., 2002). However, vertical diffusion is small in comparison to the fast sinking rates of
aggregated phytoplankton exported from the mixed layer. Therefore vertical diffusion in and
![Page 30: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/30.jpg)
Modelling Ocean Fertilisation Patch Dynamics Methodology
22
out of the mixed layer is considered negligible. There is no resuspension of particles after they
have been exported.
Diffusion in the horizontal is assumed to be important (Boyd et al., 2002). A spatially
constant length scale dependent diffusion coefficient is applied given by Kh= α L4/3 (Okubo,
1971).
Growth is nutrient-limited according to the Monod equation:
+=
SNN
maxµµ
where N is the total available nutrient, µmax is the maximum growth rate and S is the half-
saturation constant. Nutrients are assumed to be distributed by similar processes to
phytoplankton, diffusing horizontally and removed due to uptake by growing phytoplankton
according to an uptake ratio C:nutrients (eg C:Fe). Vertical diffusion of nutrients outside the
mixed layer is neglected but within the mixed layer the concentration is assumed to be
vertically uniform.
In both IronEx II and SOIREE, a single species has been shown to be dominant following
iron fertilisation (Jackson and Lochmann, 1992). Consequently, this model takes a single
species approach. This also maintains simplicity in the model by necessitating that only one
growth rate aggregation threshold is chosen.
3.2 Application of model theory The model is coded in MATLAB as diffgrow2 (see Appendix 1) and model inputs shown in
Table 3.1. Some of these inputs have been added to the model developed by Waite and
Johnson (2003), however, these affect peripheral behaviour of the model only and neither the
phytoplankton nor export dynamics.
The model outputs are shown in Table 3.2. Some of these outputs are additional to those of
the basic Waite and Johnson model. These have been incorporated into the model to identify
aspects of the nutrient dynamics (for example, tNdeplete, the time to nutrient depletion)
whereas others show phytoplankton dynamics by simply outputting what is already
determined in the model itself. This was necessary to understand how the phytoplankton
growth over time corresponds to export changes.
![Page 31: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/31.jpg)
Modelling Ocean Fertilisation Patch Dynamics Methodology
23
Table 3.1 – Inputs into MATLAB code, “diffgrow2.m” Model inputs Description
C Initial phytoplankton concentration (in terms of carbon)
Nf Fertilising nutrient distribution
Nb Background nutrient
K Horizontal diffusion coefficient
G 3 element growth vector G(1) – growth rate, µ
G(2) – half-saturation constant, S
G(3) – nutrient uptake rate (percent nutrient usage per unit growth).
T Threshold for export. Equivalent to Ccr
C=0 if C>T
dx Grid size (square matrix)
dt Time step size
steps Number of time steps
bound Defines the boundary condition 0=open
1=closed
2=periodic
movname Name of animation, if generated
p Defines plotting and animation condition
0=no plotting or animation
1=plotting and animation
Table 3.2 – Outputs from MATLAB model, “diffgrow2.m”
Model outputs Description
Cout Final surface phytoplankton distribution
Nout Final surface nutrient distribution
Eout Spatial export distribution
Et Export time series
tNdeplete Time to nutrient depletion
Cc Biomass time series
Cpua Concentration per unit area time series
3.3 Investigation of parameters Investigation of parameters was conducted by varying the values of a number of inputs and
observing the time series of export and total export generated. A basic parameter set was
![Page 32: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/32.jpg)
Modelling Ocean Fertilisation Patch Dynamics Methodology
24
determined and the values of parameters not intentionally varied were maintained at this
setting.
The parameters varied were maximum growth rate, µmax, the concentration of the nutrients
applied, f, the initial distribution of phytoplankton and the concentration distribution of
applied nutrients. Batch codes were created that took the mean of 10 passes of the model and
sampled the variation due to the random initial phytoplankton distribution.
3.3.1 Basic parameter set
The basic input parameters adopted were those used by Waite and Johnson (2003). The half
saturation constant in the Monod equation (G(2) input into the model) is set at S=0.2
µmol.Fe.m-3 and the maximum growth rate is 0.4 day-1. The uptake ratio for C:Fe is given as
0.002 by Waite and Johnson (2003), however, this is in error, and should be set at 0.2
(Johnson, pers. comm.). An uptake ratio of 0.2 is considered throughout all model runs,
however such a value is ad hoc in nature because other nutrients are likely to become limiting
when iron is in excess (Hutchins and Bruland, 1998). The threshold, or Ccr, value used
throughout all model runs was 14 mmol C m-3. This is used by Waite and Johnson (2003) but
is also supported by field samples from the subarctic Pacific and the Ross Sea, Antarctica
(Jackson and Lochmann, 1992).
The initial phytoplankton concentration, C, was given by a random concentration distribution
over the 5L by 5L domain of the model. A square patch of nutrients was seeded on the centre
of this background over an area of L by L. An L=10 cells resolution was used to represent
these length scales in the model (i.e. dx =L/10).
The timestep dt was made to be dependent upon the grid size, dx, by taking the minimum
value of either 0.2 or 0.5(dx)2. This methodology is used by Waite and Johnson (2003). The
horizontal diffusion coefficient, Kh is also length scale dependent according to the fertilizing
patch length scale, L as Kh= α L4/3 with α= 0.08 km2/3day-1 (Okubo, 1971)
3.3.2 Variation of growth and fertilising concentration
Values of maximum growth rate, µmax, were varied over the range encountered in natural
oceanic waters. Fertilising concentration, f, was more broadly varied near the range used in
the in-situ experiments conducted to date.
![Page 33: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/33.jpg)
Modelling Ocean Fertilisation Patch Dynamics Methodology
25
3.3.3 Variation of spatial distribution
As the spatial variation was deemed to be an important factor leading to high concentrations
and therefore carbon export, several patterns of initial phytoplankton distribution and
fertilised patch were trialled. These variations are demonstrated in Figure 3.1 and 3.2. All
spatial variations retain the same fertilising length scale, L, (area of enrichment is L by L) and
the same initial phytoplankton background domain of 5L by 5L.
Initial phytoplankton distribution has four cases, length scales of randomness - of L/10, L/5
and L/2 - and a deterministic case in which the concentration is uniformly applied (Figure
3.1). In random distributions, cell values are randomly generated in MATLAB with a
maximum of 6.0 mmol C m-3 and a mean of 3.0 mmol C m-3. The deterministic case applies
concentrations of 3.0 mmol C m-3. These initial phytoplankton variants will be referred to as
the randomness length scales of L/10, L/5, L/2 and the deterministic case.
Three fertilising concentration distributions were applied (Figure 3.2). Each of these had a
fertilising patch length of L but two had subpatches of lengths L/10 and L/5 (Figure 3.2 (b)
and (c)). The variants of fertilising patch distribution will be referred to as uniform, subpatch
length L/10 and subpatch length L/5 ordered as shown in Figure 3.2. In order to allow
comparison of total export between different patch variants, the overall nutrient concentration
applied over the domain was kept constant. This meant that the concentration per unit area in
some cells was increased. Applied nutrient concentration was 2.44 times higher in nutrient
cells for the L/10 case and 6.25 times higher for the L/5 case.
![Page 34: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/34.jpg)
Modelling Ocean Fertilisation Patch Dynamics Methodology
26
3.3.4 Multiple fertilisation events
Temporal variations in the way nutrients were applied were investigated, in particular, the
total export and export time series, by applying nutrients to the domain more than once.
Multiple applications of iron were a feature of both IronEx II and SOIREE.
A further function which encompassed the diffgrow model in a user-specified repeat
procedure was created (coding in Appendix 2). The domain was repeatedly fertilised with the
same fertilising patch variant over an 18 day period, followed by a period of 15 days ‘cooling
down’ without fertilisation. The period between nutrient applications was varied from 1 day
Figure 3.1 – Example initial phytoplankton distributions at the different random length scales input - a) L/10, b) L/5, c) L/2 and deterministic distribution d) uniform. The basic passes of the model use the distribution shown in a).
a
dc
b
Figure 3.2 – Nutrient distribution of fertilising patch variations – length scales a) uniform, b) L/10, c) L/5. Green indicates the presence of nutrients and blue the absence. The basic passes of the model use the distribution shown in a).
L a cb
![Page 35: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/35.jpg)
Modelling Ocean Fertilisation Patch Dynamics Methodology
27
to 2 days, 3 days, 6 days, 9 days and once in the 18 day period. It was necessary to choose
integer factors of the 18 day fertilisation period so that each fertilisation period could fit
wholly into the 18 day period. In order to allow comparison of total export the same amount
of nutrients needed to be applied overt the 18 day fertilisation period. Therefore, the total
nutrients applied at each fertilisation event was made to be a factor of the fertilisation
frequency. When fertilising once every day the factor was times1, and when fertilising once
over the 18 days the factor was times 18.
Spectral density estimates (using MATLAB’s in-built PSD function) were carried out on the
export time series output of the 10-pass ensemble of these multiple seeding runs to identify
the dominant frequencies present. Spectra were also calculated for an ensemble of 10 passes
of a longer 180 day run from which periodicity was expected to be more evident. To identify
only the dominant periods only spectral powers greater than 1x104 were considered.
![Page 36: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/36.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
28
Chapter 4 Results
4.1 Variations of growth and fertilising concentration parameters Export per unit area increases both with increasing fertilising concentration and increasing
maximum growth rate (Figure 4.1). Export increases linearly with applied nutrient
concentration, but is non-linearly affected by maximum growth rate. This means that the
effect of increasing growth rate is greatest when growth rate is small and tapers off when
growth rate is larger.
Figure 4.1 – The effect of variations of maximum growth rate, µmax, (on axis from 0 to 1) and fertilising concentration (on axis from 0 to 10) for the three different patch lengths shown. The plots represent, on the z axis, export per unit area (mmol C m-2) in the left column, time to nutrient depletion, tN (days) in the central column and tN x µmax in the right column. Derived from 10-run ensemble.
Export /area (mmol C m–2) tN (days) tN x µ
10km
25km
50km
Fertilising conc. µM/m3
µmax (/day) µmax (/day)
Fertilising conc. µM/m3
µmax (/day)
![Page 37: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/37.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
29
Increasing the patch size is also important for the amount of export generated. The frontier for
export to occur moves towards lower concentrations and lower growth rates as patch size
increases. That is, export occurs for a greater range of growth and fertilising concentration
parameters at larger length scales than it does at smaller lengths scales.
By normalising the export by the fertilising concentration, export still increases with
fertilising concentration, however the curvature of the export slope with fertilising
concentration is no longer linear. Instead it shows concentration increase to be more important
when concentration is low than when it is high.
4.1.1 Time scales of export
Figures 4.5 and 4.1 demonstrate that time to first dump and time to nutrient depletion are
unaffected by changes in fertilised concentration. These parameters are significantly affected
by growth rate and appear to be bound by an inverse relationship (Figures 4.1 and 4.4). This
means that the inverse of growth rate can be considered an important timescale in export. This
proposal is partly supported by Figure 4.1 which demonstrates a somewhat constant
relationship between the maximum growth rate, µmax multiplied by time to nutrient depletion,
tN, and maximum growth rate. An inverse relationship between µmax and tN would lead to a
constant relationship between µmaxtN and µmax, however, Figure 4.1 does not show an entirely
constant relationship. Deviations from this pattern occur when growth rate is large and where
fertilising concentration is large and these are propagated to lower maximum growth rates and
lower fertilising concentrations with increasing patch length scale.
Figure 4.2 – Export per unit area per unit concentration of nutrient applied, generated by variations of maximum growth rate, µmax (on axis from 0 to 1) and fertilising concentration (on the axis from 0 to 10). Patch length is a) 10 km, b) 25 km and c) 50km.
Exp
ort
(mm
olC
m-2
µm
olFe
-1 m
3 )
a b c
![Page 38: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/38.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
30
4.1.2 Maximum growth rate
As maximum growth rate increases the time until the first export event reduces. There is a
significant reduction from approximately 10 days, when export begins to occur, to close to 2
days at maximum growth rates of 0.7 /day. The weighted average of export flux moves
closely with the time to first export since the duration of the export event is short and
reasonably constant.
At higher growth rates there is evidence of two export peaks. The maximum export flux
increases with growth rate but there is also a secondary event with a smaller export flux. This
secondary event usually occurs after the maximum export event, however it may also occur
before (Figure 4.3e)).
a c b
d e f
Figure 4.3 – Ten-run ensemble of time series of export showing variations with maximum growth rate. Maximum growth rates a) – f) are 0.2, 0.3, 0.4, 0.5, 0.6 and 0.7 /day respectively. The fertilising concentration is 4.0 µmol m-3. Patch length, L = 25 km.
Expo
rt (m
mol
C m
-2 day
-1)
Time since nutrient fertilisation (days)
![Page 39: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/39.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
31
Figure 4.4 demonstrates the behaviour of the export time series with changing maximum
growth rate by considering some descriptive parameters. The time of the first export is related
to maximum growth rate in an apparently inverse relationship. The mean time of export,
weighted by export flux closely resembles this pattern, because as shown, export occurs
reasonably instantaneously and only increases slightly with µmax. This increase is due to the
multiple peaking demonstrated in the complete time series (Figure 4.3).
4.1.3 Fertilising concentration
Increasing fertilising concentration has little impact on the time scales involved in the export
time series (Fig 4.5). For the maximum growth rate of 0.4 day--1 used, the timing of the first
export event falls consistently about 4.2 days. The fertilising time scale does, however, appear
to have a more significant impact on the distinction of export ‘events’ that occur. Two events
are clearly distinguished for concentrations of 6.0 and 7.0 µM m-3 that are far more distinct
than the multi-peaking seen in Figure 4.3 for high growth rates. Less distinct multi-peaking of
the kind seen in Figure 4.3 are, however, evident in both the 4.0 and 5.0 µM m-3 and also
within the first and second distinguishable peaks. The multi-peaking at high fertilising
Figure 4.4 – Three parameters of export time series behaviour and how they change with increasing maximum growth rate. Weighted average is the mean time of export weighted by export flux.
0123456789
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Maximum growth rate (/day)
Tim
e (d
ays)
Weighted average Time of first export Export duration
![Page 40: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/40.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
32
concentrations is distinguishable due to the long duration of both peaks. This means that
although the initial export timing is unaffected by changes in fertilising concentration the
weighted average of export occurs more than 1 day later for the 7 µM m-3 than it does for the
3 µM –3 case. In contrast to the multi-peaking of short duration (Figure 4.5 c) and d)), the
distinguishable peaks are of relatively even export, indicating two major export events in their
own right. The characteristics of the second export event are of an increasing export flux
leading to the second peak. Animations created by the model show phytoplankton export
occurs at peripheral concentrated nodes, before the central area removed by the first export
event has recovered sufficiently from growth and diffusion from other areas to be at the
critical export concentration again, thus leading to the second export flux peak. The maximum
export flux of the time series is not substantial and if anything reduces as fertilising
concentration increases. Since export is sustained for a greater duration, however, the total
exported increases linearly as demonstrated in Figure 4.1.
a c b
d e f
Figure 4.5– Ten-run ensemble of time series of export showing variations with fertilising concentration. Fertilising concentrations a) – f) are 2.0, 3.0, 4.0, 5.0, 6.0, and 7.0 µmol m-3 respectively. The maximum growth rate is 0.4 /day. Patch length, L = 25 km.
Exp
ort (
mm
ol C
m-2 d
ay-1
)
Time since nutrient fertilisation (days)
![Page 41: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/41.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
33
Figure 4.6 demonstrates that after fertilising concentration is significantly large enough to
generate export, at 2 µmol m-3, it has little effect on the timing of the first export. There is,
however, a significant jump in the export duration at 6 µmol m-3 when the distinguishable
peaks in export begin to occur. This causes the weighted average to increase.
4.1.4 Variation of the initial phytoplankton distribution length scale of randomness
The random length scale of initial phytoplankton distribution does not cause any obvious
patterns of difference for export time series at smaller length scales, however as length scales
increase more export events occur (Figures 4.7 - 4.10). At 25km, single peaks occur for all
length scales.
Figure 4.6 - Three parameters of export time series behaviour and how they change with increasing fertilising concentration Weighted average is the mean time of export weighted by export flux.
0
1
2
3
4
5
6
0 2 4 6 8
Fertilising concentration (umol m^-3)
Tim
e (d
ays)
Weighted average Time to first export Export duration
![Page 42: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/42.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
34
Figure 4.7 – Patch length 25 km, 10-run ensemble of export time series for four length scales of randomness in initial phytoplankton distribution. Random length scale is (a) L/10, (b) L/5, (c) L/2 and (d) uniform.
a
c
b
d
Time since nutrient fertilisation (days)
Exp
ort (
mm
ol C
m-2
day
-1)
Figure 4.8 – Patch length 40 km, 10-run ensemble of export time series for four length scales of randomness in initial phytoplankton distribution. Random length scale is (a) L/10, (b) L/5, (c) L/2 and (d) uniform.
a
c
b
d
Time since nutrient fertilisation (days)
Expo
rt (m
mol
C m
-2 d
ay-1
)
Figure 4.9 – Patch length 70 km, 10-run ensemble of export time series for four length scales of randomness in initial phytoplankton distribution. Randomlength scale is (a) L/10, (b) L/5, (c) L/2 and (d) uniform.
a
c
b
d
Time since nutrient fertilisation (days)
Exp
ort (
mm
ol C
m-2
day
-1)
Figure 4.10 – Patch length 85 km, 10-run ensemble of export time series for four length scales of randomness ininitial phytoplankton distribution. Random length scale is (a) L/10, (b) L/5, (c) L/2 and (d) uniform.
a
c
b
d
Time since nutrient fertilisation (days)
Exp
ort (
mm
ol C
m-2
day
-1)
![Page 43: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/43.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
35
The number of export peaks occurring in each of the fertilising length scales is summarised in
Table 4.1. It shows that the number of peaks increases as the patch length scale increases but
this then decreases at the largest length, 85km.
Table 4.1 – Number of export peaks in 10-run time series ensemble.
Growth rate is 0.4 /day.
L/10 L/5 L/2 Uniform
25 km 1 1 1 1
40 km 2 2 3 1
70 km 3 3 4 2
85 km 1 2 3 1
The timing of the end of export is constant for each of the background phytoplankton
distributions, however, the duration of export is greater when the randomness length scale is
L/2 than in other cases. Consequently the time of the first export is earlier in the case of the
L/2 randomness length scales distribution.
Figure 4.9 demonstrates that there is very little difference between the amounts of export per
unit area generated by any of the different randomness length scales. The export per unit area
increases with patch length at the same rate for each of the random phytoplankton distribution
length scales, however there is a great deal of variation about this trend.
Figure 4.11 – Export per unit area generated over 10 days from a single fertilisation for different length scales of randomness of the initial phytoplankton distribution.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 20 40 60 80 100
Patch length (km)
Expo
rt (m
mol
C m
^-2)
length L/10
length L/5
length L/2
uniform
![Page 44: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/44.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
36
4.1.5 Variation of the fertilising concentration length scale
Figures 4.12-4.15 show that the uniform application of nutrients mainly generates single
peaks. By contrast, the smaller subpatch length scales are associated with multiple peaks at all
patch length scales. The maximum export flux of the different distributions is greatest at the
smallest (L/10) length scale. The L/5 length scale generally has a smaller maximum export
flux than either of the other distributions but has a greater duration of export than the uniform
nutrient application and equal or greater duration than the L/10 length. Note that the scales of
the ‘export’ axis vary between Figures 4.12 to 4.15 (but not within a figure).
The length scale of the fertilising subpatches has a significant effect on the export per unit
area (Figure 4.16). There is greatest divergence between the exports generated for different
fertilising distributions when the fertilising length scale is large.
Figure 4.12 – Patch length 25 km, 10-run ensemble of export time series for three length scales of fertilising subpatch application. Length scale is (a)uniform, (b) L/5 and (c) L/10.
a
c
b
Time since nutrient fertilisation (days)
Expo
rt (m
mol
C m
-2 d
ay-1
)
Figure 4.13 – Patch length 40 km, 10-run ensemble of export time series for three length scales of fertilising subpatch application. Length scale is (a) uniform, (b) L/5 and (c) L/10.
a
c
b
Time since nutrient fertilisation (days)
Expo
rt (m
mol
C m
-2 d
ay-1
)
![Page 45: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/45.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
37
Figure 4.14 – Patch length 70 km, 10-run ensemble of export time series for three length scales of fertilising subpatch application. Length scale is (a) uniform, (b) L/5 and (c) L/10.
a
c
b
Time since nutrient fertilisation (days)
Exp
ort (
mm
ol C
m-2
day
-1)
Figure 4.15 – Patch length 85 km, 10-run ensemble of export time series for three length scales of fertilising subpatch application. Length scale is (a) uniform, (b) L/5 and (c) L/10.
a
c
b
Time since nutrient fertilisation (days)
Exp
ort (
mm
ol C
m-2
day
-1)
0
0.05
0.1
0.15
0.2
0.25
0 20 40 60 80 100
Patch length (km)
Expo
rt (m
mol
C m
^-2)
uniform
length L/5
length L/10
Figure 4.16 – Export per unit area generated over 10 days from a single fertilisation for different length scales of fertilising nutrient subpatch. Note significant divergence between length scales.
![Page 46: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/46.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
38
0
50
100
150
200
250
300
350
400
450
0 50 100
Patch length, L (km)
Max
imum
exp
ort f
lux
(mm
ol C
m^-
2 da
y^-1
)
BasicRandomness L/5Randomness L/2DeterministicSubpatch L/5Subpatch L/10
0
1
2
3
4
5
6
7
0 20 40 60 80 100
Patch length, L (km)
Mea
n ex
port
tim
e w
eigh
ted
by e
xpor
t flu
x (d
ays)
BasicRandomness L/5Randomness L/2DeterministicSubpatch L/5Subpatch L/10
Figure 4.17 – Maximum export flux occurring in different spatial scales of initial phytoplankton distribution and fertilising subpatch scale and how these vary with the length scale, L.
Figure 4.18 – Mean time of export weighted by export flux occurring in different spatial scales of initial phytoplankton distribution and fertilising subpatch scale and how these vary with the length scale, L.
![Page 47: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/47.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
39
0
1
2
3
4
5
6
0 20 40 60 80 100
Patch length, L (km)
Tim
e to
the
first
exp
ort (
days
)
BasicRandomness L/5Randomness L/2DeterministicSubpatch L/5Subpatch L/10
0
0.5
1
1.5
2
2.5
3
0 20 40 60 80 100
Patch length, L (km)
Expo
rt d
urat
ion
(day
s)
BasicRandomness L/5Randomness L/2DeterministicSubpatch L/5Subpatch L/10
The greatest export flux at most L occurs at the smallest, L/10 subpatch length scale (Figure
4.17). This is in addition to generating the greatest export per unit area (Figure 4.16) The
deterministic, uniform initial phytoplankton distribution also demonstrates large export flux
associated with simultaneous sinking of the entire patch which occurs, as Figure 4.20 shows,
over small durations relative to the rest of the spatial distributions. The duration of export
does not increase greatly even as the length scale increases for the deterministic case,
however, in most other spatial variants there is an increases with the length scale, and in the
case of the L/5 scale subpatch, a significant jump in the export duration that coincides with
Figure 4.19 – Timing of the first export occurring in different spatial scales of initial phytoplankton distribution and fertilising subpatch scale and how these vary with the length scale, L.
Figure 4.20 – Timing of the first export occurring in different spatial scales of initial phytoplankton distribution and fertilising subpatch scale and how these vary with the length scale, L.
![Page 48: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/48.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
40
temporally distant multiple export events (Figures 4.12-4.15). The export duration is
considerably affected by the spatial variations. By contrast, the spatial variations do not have
a significant impact on the timing of the first export (Figure 4.19). These remain constant with
the length scale L also, suggesting that the length scale does not affect this timescale. The
exception to this is in the largest random length scale, L/2 which is variable, but does not
exhibit any discernible relationship with L. Figures 4.17 to 4.20 demonstrate that export does
not occur for L=10km and the fertilising concentration of 3 µmol m-3 and maximum growth
rate of 0.4 day-1 used throughout. This is with the exception of the smallest, L/10 subpatch
length scale of fertilising concentration distribution, which does export, demonstrating that
this variant is capable of supporting export at a wider range of parameters.
4.1.6 Multiple fertilisation events
During multiple fertilisations, the amount exported from the mixed layer increases as
fertilisations become less frequent (Figure 4.21). It is likely that the effect of the amount of
nutrients applied at each fertilisation which is coupled to how often fertilisations occur also
has a significant impact on the greater amount exported. Q has little effect on the amount
exported by comparison to the effect of the inter-fertilising period.
Figure 4.21 demonstrates that the same relationship between fertilising period and Q exists for
each of the fertilising patch distributions. Export is, however, greater as fertilising patch
length scales decrease. This effect is most pronounced for large fertilising periods.
Although Q appears to have very little effect on export for values between 0.01 and 0.05, at
larger values of Q export decreases rapidly. There is an especially large downturn at large
fertilising periods (Figure 4.17). This demonstrates that when Q is small, maximum export is
greatly favoured by large fertilising periods, however, when Q is large, maximum export is
only marginally increased by applying greater fertilising periods and the associated increase
in fertilising concentration.
![Page 49: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/49.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
41
4.1.7 Multiple fertilisation export time series
The export time series produced by multiple fertilisation events are complex and appear
highly random (Figure 4.23 and 4.24). There is not clear evidence of regular periods of
export, for any of the fertilising frequencies, in the 18 day pass, however, regularity of export
is most obvious when the patch length is small (e.g. – 10km). At small length scales, export
Figure 4.22 – Export per unit area generated by the uniform fertilising distribution. Q varied by µmax. Note that the range of Q, especially where the downturn occurs, is not shown in Figure 4.21.
Figure 4.21 – Export per unit area generated by three fertilising distributions – at length scales of L/10 and L/5 – and uniform. Q varied by patch length, L. µmax = 0.4 /d.
L/10 L/5
Uniform
![Page 50: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/50.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
42
flux variation is large in comparison to larger patch lengths (80km) where the export flux is
smaller but more consistent. At inter-fertilisation periods of 9 days it is possible to see the
distinctive patterns of individual fertilisation events as export ceases and then resumes
following fertilisation. Export flux is often greatest when nutrients are reapplied following a
period of no export.
Figure 4.23 – Ten-run ensemble of export time series for multiple fertilisations at different patch length scales corresponding to Q = 0.0431, 0.0271, 0.0207, 0.0171, 0.0147, 0.0130, 0.0118, 0.0108 respectively. Growth rate is 0.4 /day. The period between fertilisations is 1 day (upper left), 3 days (above) and 9 days (left) and is indicated by the dotted line.
10km
20km
30km
40km
50km
60km
70km
80km
10km
20km
30km
40km
50km
60km
70km
80km
10km
20km
30km
40km
50km
60km
70km
80km
![Page 51: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/51.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
43
Variations of µmax clearly alter the time scales in the export time series, most obviously the
time to the first export event (Figure 4.24). Large export flux spikes are observed in low Q
values when µmax is varied, in contrast to the same being observed in high Q when length
scale is varied. At high Q (low µmax) export is still variable but the amplitude of export flux is
smaller. Export peaks are evident following re-fertilisation after a period of no export.
Figure 4.24 – Ten-run ensemble of export time series for multiple fertilisations at different patch maximum growth rates corresponding to Q = 0.0936, 0.0624, 0.0468, 0.0374, 0.0312, 0.0267, 0.0234, 0.0208, 0.0187, 0.0170 and 0.0156 respectively. Patch length is 25km. The period between fertilisations is 1 day (upper left), 3 days (above) and 9 days (left) and is indicated by the dotted line.
0.15 /d
0.1 /d
0.2 /d
0.25 /d
0.3 /d
0.35 /d
0.4 /d
0.45 /d
0.55 /d
0.6 /d
0.5 /d
0.15 /d
0.1 /d
0.2 /d
0.25 /d
0.3 /d
0.35 /d
0.4 /d
0.45 /d
0.55 /d
0.6 /d
0.5 /d
0.15 /d
0.1 /d
0.2 /d
0.25 /d
0.3 /d
0.35 /d
0.4 /d
0.45 /d
0.55 /d
0.6 /d
0.5 /d
![Page 52: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/52.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
44
Power spectral analysis of a 180 day re-fertilisation length reveals dominant periods of export
at a variety of values. These can be seen in Figure 4.25, and are summarised from closer
investigation, in Table 4.2. The dominant period coincides with the fertilising period,
however, other periods are also important. Multiples of the forcing period are evident. Export
periodicity of approximately 2 days is evident in Figure 4.25 at all fertilising periods.
Powerful periods are densely distributed at lower values. The dominant export periods that are
evident for each of the different fertilising periods are summarised in Table 4.2.
When the fertilising period is 1 day the dominant export periods are approximately 1.5 days
and 0.5 days. Figure 4.25 demonstrates how the fertilising periodicity represents a period of
low power, not high, at fertilizing periods of 1.
Figure 4.26 demonstrates the effect of the patch length on the dominance of certain periods,
for the 6 day inter-fertilising period. The 6 day fertilising period is important but generally of
equalled in dominance by the 2 day period at all patch sizes. Important periods between 0.2
days and 2 days that are evident at the 10km patch length are not important as the length scale
1 day
3 days
6 days
9 days
18 days
2 days
Figure 4.25 – Power spectrum density of export time series of 180 day pass. Analysis of final 160 days. The 6 fertilising periods are shown. Power is on a logarithmic scale.
![Page 53: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/53.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
45
increases. For small length scales, a greater variety of small periods are dominant in the time
series, in comparison to the larger patch lengths. The lower extent of these periods exists at
approximately 0.2 days, corresponding to the timestep, dt.
Table 4.2 – Distinguishable periods of export from spectral analysis of export time series.
Fertilising period (days) Important export periods (days) 1 1.5 2 2 3 1.5, 3 6 1, 1.5, 2, 3, 6 9 1, 1.125, 1.25, 1.5, 1.7, 2.25, 3, 4.5, 9 18 Several small periods, 1.5, 1.6, 1.75, 2, 2.25, 2.6, 3, 3.6, 4.5, 6, 9, ~18
The spectral density plots of each of the spatial distributions of fertiliser application (Figure
4.27) do not show consistent periods of export between them. Phytoplankton initially applied
uniformly at a specified concentration has the most numerous and powerful period peaks.
Periods of 0.5 and 2 days appear dominant in many cases.
10 km
40 km
50 km
60 km
70 km
80 km
30 km
20 km
Figure 4.26 - Power spectrum density of export time series of 180 day pass. Analysis of final 160 days. 8 patch lengths for a fertilising period of 6 days. Power is on a logarithmic scale.
![Page 54: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/54.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
46
Basic
Phyt.
Fert. L/5
Phyt.
Phyt.
Fert.
b
Basic
Phyt.
Phyt.
Phyt.
Fert. L/5
Fert.
c
Phyt.
Phyt.
Fert.
Basic
Phyt.
Fert. L/5
d
Basic
Phyt.
Phyt.
a
Phyt.
Fert. L/5
Fert
Figure 4.27 - Power spectrum density of export time series of 18 day pass of different spatial distributions. Power is on a logarithmic scale with lower cut-off at 10,000. ‘Basic’ represents phytoplankton randomness at a length scale of L/10 and uniform fertilising distribution. ‘Phyt. L/5’ represents initial phytoplankton randomness at a length scale of L/5 and uniform fertilising distribution. ‘Phyto. L/2’ represents initial phytoplankton randomness at a length scale of L/2 and uniform fertilising distribution. ‘Phyt. deterministic’ represents uniform initial phytoplankton distribution and uniform fertilising distribution. ‘Fert. L/5’ represents initial phytoplankton randomness at a length scale of L/10 and fertilising distribution at sub-patch length scale of L/5. ‘Fert. L/10’ represents initial phytoplankton randomness at a length scale of L/10 and fertilising distribution at sub-patch length scale of L/10. a) – f) are for fertilising periods of 1, 2, 3, 6, 9 and 18 respectively. Patch length, L = 40km, µmax=0.4 /d.
![Page 55: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/55.jpg)
Modelling Ocean Fertilisation Patch Dynamics Results
47
Basic
Phyt.
Phyt.
Phyt.
Fert. L/5
Fert.
e
Basic
Phyt.
Phyt.
Phyt.
Fert. L/5
Fert.
f
Figure 4.27 (continued) -
![Page 56: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/56.jpg)
Modelling Ocean Fertilisation Patch Dynamics Discussion
48
Chapter 5 Discussion
5.1 Interpretation of results
5.1.1 Variations of maximum growth rate and fertilising concentration
The optimal conditions for export of carbon below the mixed layer are shown to occur in this
model at high fertilising concentrations and high maximum growth rates. However, neither of
these parameters can be considered to directly affect export in any particular cell. At each cell
the amount exported in any one event must be Ccr, the critical threshold at which aggregation
is specified to occur, regardless of how fast the given µmax enables Ccr to be reached.
Likewise, the fertilising concentration, f, will not directly affect how much is exported at any
particular cell in one event, because the ratio of nutrient uptake remains constant and thus the
concentration Ccr corresponds to a constant amount of nutrients, Ncr.
The method by which these parameters have the impact on export shown, is therefore not by
how much export is generated at one spot at one time, but rather how many of these areas can
be made to reach this critical export threshold, Ccr and also how areas that have already
exported once in the run, can do so again. Increasing µmax and f enables this to occur by
sustaining a high nutrient-limited growth rate, µ(t) at many cells for a large amount of time,
where µ(t) is the Monod growth rate given in Chapter 3. Consequently, growth increases and
diffusive inputs are larger than diffusive losses at as many cells as possible throughout the
length of the run. This means that Ccr can be reached in as many times in as many places as
possible, resulting in the most efficient export for the amount of nutrients supplied.
There is a constant increase in export with both µmax and fertilising concentration, f, because
this enables growth increases to dominate over diffusive losses in more of the patch and
generate greater total export fluxes.
One of the other ways that the efficient use of nutrients is manifested is in the pattern of
‘multiple peaking’. Multiple export peaks are shown to occur in the export time series at the
same large values of µmax and f at which the maximum export occurs. It can be speculated that
![Page 57: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/57.jpg)
Modelling Ocean Fertilisation Patch Dynamics Discussion
49
the processes by which multiple peaks occur as µmax increases will be in sustaining large µ(t)
in space, whereas multiple peaks as f increases can be attributed to sustaining large µ(t) over
time. These spatial and temporal processes will, of course, work in concert. That is, as both
µmax and f increase, µ(t) will be sufficient to promote growth increases over diffusive losses in
space as well as over time.
This idea can be illustrated by considering an export time series that occurs when either µmax
or f is large. Firstly consider µmax.
When µmax is large, phytoplankton in the centre of the patch will grow rapidly and quickly
attain the concentration threshold for export. This results in the first export event. The timing
of this first export event is dependent primarily upon µmax. This is because the realised growth
rate is not initially nutrient-limited, nor are the diffusive losses substantial because the
concentration gradients in the centre of the patch are small. By contrast, at the edges of the
patch there are significant nutrient and phytoplankton concentration gradients between the
nutrient enriched patch and the surrounding waters. This results in significant losses of both
nutrients and phytoplankton to the surrounding waters. Therefore growth rates are highly
nutrient-limited and a large proportion of the phytoplankton that does grow is lost across the
diffusive gradients. When µmax is large, however, these losses are less than the overall
increases in phytoplankton concentration from growth. The concentrations of phytoplankton
therefore reach threshold concentrations for aggregation and export, resulting in a second
export event. The timing of this second event is delayed because unlike in the centre of the
patch, diffusive losses of both nutrients and phytoplankton reduce the net growth rates.
In the case that f is large, phytoplankton concentrations again increase in the centre of the
patch faster than they do at the edges. However, the speed at which they grow in the centre is
in this case not dependent upon f. When f is even reasonably large, the nutrients available will
not be greatly limiting and therefore µmax, and not f, remains the best predictor of the timing of
the first export event. At the edges of the patch, the large concentration gradients again result
in losses from the patch to the surrounding waters. For intermediate values of µmax, these
losses dominate over the growth. Nutrients are lost at a rate greater than they can be taken up.
Consequently, no export occurs at the edges of the patch. In the centre of the patch, however,
this first export event has vacated areas of the patch for which the phytoplankton
concentration has returned to zero. Since f is large, however, there remains a significant
nutrient base in the vacated centre of the patch. The realised growth rate, µ(t) therefore
![Page 58: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/58.jpg)
Modelling Ocean Fertilisation Patch Dynamics Discussion
50
remains large and the diffusive gradient setup initially between the rim and the empty centre
of the patch favours input rather than loss of phytoplankton. Consequently, growth increases
of phytoplankton dominate in the centre of the patch and this area once again reaches the
export threshold. This second export event now occurs some time after the first and is
characterised by an increasing export flux prior to the peak, probably as areas closer and
closer to the centre reach Ccr. It can be seen that as f is increased the duration of the entire
export series increases.
If µmax and f are both large enough there will be evidence of these processes operating
together and delivering several peaks corresponding to export at different distances from the
centre of the patch, as Figure 4.5 f. shows.
Diffusion has been mentioned as an important part of the spatial phytoplankton dynamics. It
can be assumed that concentration gradients of the same order will exist in any 10-run
ensemble of a specified combination of µmax and f. Therefore by increasing L we are
increasing Kh and the effective contribution that diffusive processes make to the spatial
dynamics of the phytoplankton patch. At the edges of the patch this will increase losses,
however in the rest of the patch, this aids in how rapidly phytoplankton concentrations return
to threshold levels. Overall, this study shows that increasing the length scale, L, increases the
export per unit area.
5.1.2 Time scales of export
Maximum growth rate is one of only two time based parameters input to the model. Since
many of the timescales are greatly dependent upon the maximum growth rate, the inverse of
growth rate µmax-1 can be considered an important time scale. In particular, the inverse of
growth rate is correlated with the timing of the first export event and of nutrient depletion.
The time to the first export does not change greatly with the patch length, which, since Kh is
length scale dependent, suggests that Kh does not affect this timescale.
The time to nutrient depletion, tN, is not wholly dependent upon µmax. At high maximum
growth rates and fertilising concentrations, there is not a linear relationship between tN and
µmax-1. High values of µmax and f coincides both with the greatest export and the occurrence of
multiple export peaks in the export time series. Multiple export peaks continue to remove
phytoplankton from the domain not once but several times. Therefore nutrients are able to
diffuse into areas where there are no nutrients more often and hence reside for longer before
![Page 59: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/59.jpg)
Modelling Ocean Fertilisation Patch Dynamics Discussion
51
the domain is finally depleted. Due to the role that diffusion plays in this scenario, L2/Kh may
be an important timescale at high growth maximum growth rates and high fertilising
concentrations.
5.1.3 Spatial distributions
The length scale of the randomness of the initial phytoplankton distribution is used as a crude
attempt to quantify the correlation of different patch scales that exist in the open ocean. It was
suspected that due to the non-linear nature of the phytoplankton dynamics, changes in the
randomness applied may be propagated through to the export generated. Areas randomly
assigned high concentrations close together would be expected to grow quickly together,
diffusive losses would be small across small concentration gradients, and export would be
high. By comparison, smaller length scales may not necessarily have a high concentration of
phytoplankton nearby and high concentration gradients will be formed which deplete
phytoplankton stocks via diffusive losses. This makes it less likely that the critical
concentration will be reached in such areas. However, this effect is not shown to be important.
It is likely that the concentrations of the background phytoplankton are too low in comparison
to the nutrient-rich patch to have a significant effect on the patch dynamics.
There is no obvious pattern in the time series of export within or between the different
random length scales. It is likely, however, that the export behaviour becomes more variable
as we increase the length scale of randomness since the random concentrations generated, be
they high or low, occur over a larger area and thus have a greater effect on export flux than
would more localised high or low concentrations.
The distribution of the applied nutrient, by contrast, has a significant effect on the amount of
export generated. In order to generate the greatest export, it is desirable to maintain as large
an area as possible at the critical concentration for aggregation. Areas where export has
occurred reduce their concentration from Ccr to 0 within one time step, and they are of little
use for export until they return back to the Ccr range. Restoring concentrations is done firstly
by diffusion from neighbouring phytoplankton areas and secondly by growth of
phytoplankton diffused into the vacated area. Hence in order to return the exported area most
quickly back to Ccr adequate diffusion must occur. This means that ideally every exported
area will have an adjacent area of relatively high phytoplankton such that the concentration
gradients are large. Hence we desire the exported areas to be small.
![Page 60: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/60.jpg)
Modelling Ocean Fertilisation Patch Dynamics Discussion
52
In order to achieve this growth-export pattern it is necessary to promote growth in certain
areas, through the addition of fertilising nutrients, whilst leaving other areas to grow at a
different rate. This reasoning was tested by the variations in the spatial distributions of the
fertilising patch. This study shows that by decreasing the length scale of the subpatches from
a uniform application to an L/5 subpatch size, to a L/10 subpatch size increases the amount of
export generated.
No particular patterns are evident in the export time series between or within different
fertilising length scales when fertilising once. Since only one fertilising event was modelled
here, it is likely that the effect of random seeding and the non-linearity of the model cause
significant variations in the exact fluxes or the exact duration of multiple flux peaks. It is
expected that any regularity in export time series behaviour will only be observed when the
fertilising patch regime is applied repeatedly through multiple fertilising events.
5.1.4 Multiple fertilisation events
Fertilising more than once enables concentrations of phytoplankton to be maintained at or
close to the critical aggregation threshold. This study shows that longer periods between
fertilisation generate the greatest carbon export. However, in order to compare fertilising
frequencies the same amount of nutrients need to be applied for each frequency over the entire
fertilising period. This means that longer periods are also those associated with the greatest
instantaneous inputs of nutrients.
Given the importance of fertilising concentration in generating the greatest total export,
mentioned above, it is not clear whether it is the fertilising concentration or the length of time
that is responsible for export being greater at greater inter-fertilising periods. In a practical
sense, the reasons are inconsequential because the more general result, that a certain amount
of iron is best applied at higher concentrations rarely rather than at lower concentrations more
often, is valid regardless of the process.
In comparison to the effect that the inter-fertilising period plays in the export generated, the
Waite and Johnson non-dimensional parameter, Q, plays only a minor role. In considering the
most effective inter-fertilising period, it is important to note, though, that at low Q the effect
on total export of having long periods between fertilising is much greater than if the same
decision were to be made at higher values of Q.
![Page 61: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/61.jpg)
Modelling Ocean Fertilisation Patch Dynamics Discussion
53
The export time series generated during multiple fertilisations is highly complex, reflecting
the non-linearity of the model. Periodicity is evident in the 18 day passes of the model only at
the later days. Power spectral density analysis of the 180 day base case reveals distinct
periods in the time series. These periods are dominated by the forcing period of the fertiliser
application and integer multiples of these periods (e.g. – T, T/2, T/3, T/4,…,T/n). This
pattern, and the relative power at each of the periods, is evidence of a saw-tooth pattern of the
time series, for which the period is that of the fertiliser application. That is, there is an initial
abrupt export flux peak and that export flux following this decreases approximately linearly
until the next fertilising event.
The power associated with the 2 day period is greater than one would expect for an exact
sawtooth formation. It is usually of equivalent power to the forcing (fertilising) period itself.
This suggests that 2-daily export is an important period of export, at all patch lengths. The 2
day period is also the most consistently observed period in the power spectral density analysis
of the spatial distribution variations. The results of the power spectral density analysis of the
spatial distribution analysis should be considered very tentatively, however, as they were
computed from only 18 day passes and are therefore insufficient to reliably apply power
spectral analysis.
It is evident that for small length, L, the behaviour of the export time series is quite ‘peaked’.
Export fluxes are higher and then fall again regularly. As L is increased the time series is
characterised by more constant export flux. Although, as mentioned above, this does not
contribute to a substantially greater amount of total export in comparison to the effect of the
period between fertilising events, it again illustrates the importance of the role of diffusion,
mediated by the length scale, L, on the phytoplankton dynamics. A particular behaviour that
is observed for longer inter-fertilisation periods, where L is small, is an especially large export
flux immediately following the reapplication of fertiliser. This behaviour is not observed
when L becomes larger as it is an effect of the size of Kh. As L is small, Kh is small and the
nutrients become low or depleted before the next fertilising event takes place. Growth is very
nutrient-limited during this period and diffusion is important in spreading the patch more
uniformly over the domain. Therefore, when fertiliser is reapplied, a large peak in export flux
is observed as a large amount of phytoplankton simultaneously grow and sink out.
![Page 62: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/62.jpg)
Modelling Ocean Fertilisation Patch Dynamics Discussion
54
5.1.5 Summary of interpretations
Although there are a number of aspects of the results of this study analysed, the major
conclusions are easily summarised. Almost all the behaviour observed in every aspect of the
study can be considered a balance between the parameters affecting growth increases and
those affecting diffusion.
In order for the maximum export to be generated the growth rate increases need to be greater
than the losses for as much of the patch as possible for as much of the time. This can be
achieved by increasing the maximum growth rate, µmax or increasing the nutrients that are
available, by increasing f. Large µmax has an effect on export by making sure that growth
increases are greater than diffusive losses even quite close to the edges where concentration
gradients are high. This can result in multiple peaks close together as the centre and then parts
of the outer rim reach the aggregation threshold soon after one another. Large f has an effect
on export by making sure that nutrients are always readily available, thus maintaining
sufficient growth increases even when some has been removed over concentration gradients at
the edges of the patch or allowing more than one export event to occur at the centre of the
patch.
Promoting this constantly high growth increases across the domain is critical in overcoming
diffusive losses. In both time and space, this study shows it is a more efficient use of
fertilising nutrient to apply more at small time or space scales than less over larger domains.
Therefore spatial distributions of nutrients are shown to be most effective when nutrients are
applied to localised areas rather than at lower concentrations over broader areas. In the spatial
case diffusive effects can also be advantageous in reseeding areas that have been vacated by
an export event, therefore smaller sub-patch scales effectively allow diffusion to have a
greater effect as every point on the surface is close to a diffusing source of phytoplankton and
the concentration gradients set up are large.
In the time domain, it is again shown that applying nutrients at high concentrations
infrequently generates greater export than does applying nutrients frequently at low
concentrations. This is analogous to the spatial case as this regime firstly promotes rapid
growth, but secondly gives a long enough time for diffusion to act to reseed all areas that have
been vacated by export, and thus efficiently access the nutrients as they are reapplied.
![Page 63: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/63.jpg)
Modelling Ocean Fertilisation Patch Dynamics Discussion
55
Due to these simple balances, only a small number of parameters are important: µmax and the
fertilising concentration, and L. These not only affect the export generated but also the timing
of many of the export events. µmax-1 is the dominant timescale affecting the timing of the first
export, and the time until the nutrients become depleted for small fertilising concentrations
and small µmax. At larger values of these, L affects the diffusive timescale L2/Kh which is
important. The duration of the export event is correlated with how much nutrients were
available initially, i.e. – the fertilising concentration, f. That most of the time and export
behaviour of this model can be represented by simple balances between µmax, fertilising
concentration and L is an important result in easily applying the results.
Although the exact spatial dynamics of phytoplankton distribution are never the same in any
model run, if we at least assume that the internal scales of the export generated ‘holes’ are
similar for the same µmax and f then the concentration gradient is not likely to be significantly
different. This means that the diffusive processes can be considered to increase as L. Where
this reasoning does not apply is in explaining where comparisons are made for different
spatial distributions of the fertilising patch. In this case the internal length scales will be
different between different variants and quantifying these differences merely by L is not
sufficient. This also affects the consideration of a single number designed to characterise the
complete behaviour of the exporting system. Although Waite and Johnson (2003) have aptly
applied the non-dimensional parameter, Q, to a uniform fertilising patch when considering
merely the total export generated, the spatial scales on which it works make it alone,
unsuitable for representing the effect of smaller internal scales generated by more complex
fertilising patch distributions.
5.2 Practical implications
Scientific interest in the role that iron plays in the phytoplankton physiology and bloom
development needs to be set aside if iron fertilisation were to be used for reducing the impact
of global carbon emissions. The ultimate goal in using iron fertilisation for climate change
regulation must be in line with the second tenet of Martin’s iron hypothesis, to increase the
carbon export to the deep and intermediate oceans (Martin, 1990). In the deep ocean carbon
can be removed from the atmospheric carbon pool for a substantial amount of time (Wigley
and Schimel, 2000).
![Page 64: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/64.jpg)
Modelling Ocean Fertilisation Patch Dynamics Discussion
56
Waite and Johnson (2003) have shown that large patch lengths (> 10km) are necessary to
generate export for growth rates of 0.4d-1. In particular, the Southern Ocean, which offers the
greatest HNLC region in which to apply iron fertilisations, requires especially large length
scales due to the temperature-limited growth rates. Considering that such large scales of
application are necessary it is important that the efficient use of resources be considered. The
amount of iron used, the time required to carry out the fertilisation and the number of times it
is necessary to reapply iron are all factors that need to be optimised.
Many factors affect the maximum amount of export that can be generated. This study has
identified growth rate, fertilising concentration and the spatial distribution of the fertilising
patch in particular, but others have been identified in the literature, especially wind shear
which affects aggregation behaviour (Jackson and Lochmann, 1992), and zooplankton grazing
(Boyd, 2002). Only a proportion of these will, however, be able to be manipulated in order to
achieve maximum carbon export from a natural system, while others are properties of the
natural system to which the fertilising patch is applied. The most obvious is fertilising
concentration and the timing and distribution of the nutrient applied.
This study has attempted to make different variations in spatial distribution and timing of
fertilising by ensuring that over the entire length of multiple fertilisations, or over the entire
spatial domain, the total applied nutrients is constant. Consequently, this has resulted in high
concentrations needing to be applied per unit area or per unit time to balance other cases
where low concentrations are applied over a larger domain or more regularly. This presents
the following problems – 1) That the increased export observed is due to the high growth rates
created and not directly from the spatial or temporal variation, 2) That growth will actually be
limited by another nutrient as a result of high growth; and 3) That there is a practical limit to
how specific the actual time or space that nutrients are applied at is.
As mentioned previously, the first problem is a problem only for differentiating the processes
responsible. The result, that nutrients applied locally, in space or in time, generate the greatest
export remains valid regardless of which process is responsible. Unfortunately the outcome of
this first problem leads directly to the second. If the very high growth rate is responsible for
the export amounts seen, then such estimates are not practically possible.
One of the assumptions made by the model is that only one nutrient limits growth, and this is
taken to be iron when considering iron enrichment experiments. When very large growth rates
![Page 65: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/65.jpg)
Modelling Ocean Fertilisation Patch Dynamics Discussion
57
are generated in the model, this is because iron is not limiting anymore. In reality, however,
other nutrients will then become limiting. Silicate limitation has been put forward to explain
the phytoplankton dynamics, especially of the dominant diatoms, in the IronEx I experiment
(Coale et al., 1998) and carbon limitation due to carbon drawdown was also of concern in
IronEx II (Coale et al., 1996). This means that although the model predicts that large amounts
of export can be generated as the fertilising concentration is increased (up to 18µM m-3 for the
18 day inter-fertilising period) the growth with which this export is associated is not realistic.
It is more likely that, in reality, a total export peak will be observed at which the most export
is generated for increasing fertilising concentrations. After this peak, increasing the amount of
iron will do little to increase export.
This study has demonstrated that, per unit fertiliser, the greatest increase in total export occurs
when fertilising concentration is increased from small to medium values, rather than from
medium to high values. This implies that the best return for an application of iron occurs
when the first amounts are applied. At this end, iron is still limiting and therefore the models
predictions hold. Therefore the benefits of even small additions of iron should be considered.
The third problem mentioned above is one of the application of this concentrated iron
solution. Theoretically, this model predicts that decreasing the length scale of the subpatches
will lead to greater carbon export from the mixed layer. There will be a limit, however, to
what is a practicable area to fertilise.
The time at which events within the export time series occur is important for efficient
monitoring of the iron fertilisation process. Using µmax-1 needs to be investigated further, but
is an important timescale in defining these events. A parameter like L2/Kh becomes important
if the iron fertilising concentration increases also. Quantifying how these timescales affect
timing of export needs to be considered in increasing the ability to predict the behaviour of
the phytoplankton patch.
Developing a small number of parameters that combine the values µmax, fertilising
concentration, and L would be valuable in defining regimes under which certain export
behaviours would be expected to occur. The Waite and Johnson parameter, Q, was developed
to characterise the conditions for which export would occur. However, Q can best be
considered a parameter which defines bulk parameters of export rather than fine scale
variations. Other non-dimensional parameters may also need to be considered.
![Page 66: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/66.jpg)
Modelling Ocean Fertilisation Patch Dynamics Discussion
58
Clearly, in a nutrient-limited model using the Monod equation or similar, the ‘realised’
growth rate µ(t) in Q will be variable with time, and therefore cannot be used as a quantitative
measure to characterise the conditions needed for the onset of export. Choosing µmax seems to
be the logical approach to overcoming this, however, it is also clear that fertilising
concentration has considerable impact on the nutrients that are available at any point in time,
governing the realised growth, µ(t). Thus, the initial fertilising concentration needs to be taken
into consideration when formulating a bulk characterising parameter, as Q was intended to do.
This is especially true if such a parameter were to be considered when developing a fertilising
methodology to mitigate climate change. Fertilising concentration, unlike µmax is easily
manipulated, and, in addition to the timing of multiple fertilisations and the spatial
distribution of the fertilising patch, is likely to be a major factor to consider in generating the
greatest, and most efficient carbon export.
Q also incorporates the length scale, L and therefore is sensitive to changes in L. However,
the time to the first export event, which is an important timescale in managing export
behaviour, is mostly dependent upon µmax alone.
The non-dimensional parameter Q also fails to accurately consider the different length scales
at work when subpatches of fertilising concentration are applied. Q is applicable for the
uniform application of a fertilising patch, however, when subpatches exist, the scale at which
the important concentration gradients are formed are at scales significantly smaller than L.
Nevertheless, Q remains a robust parameter for the description of the bulk export in the patch.
What is needed is the development of other parameters that can be used in conjunction with Q
to better describe the spatial and temporal complexities of phytoplankton dynamics in these
optimised schemes. Examples of such parameters may be Tfertilsing vs L2/Kh for quantifying
temporal variation from multiple fertilisations (Tfertilsing is the period of these fertilisations), or
l vs L for quantifying spatial variation where l is the length of the subpatches of the fertilising
patch and L is the length of the patch as used in Q.
![Page 67: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/67.jpg)
Modelling Ocean Fertilisation Patch Dynamics Discussion
59
5.3 Issues concerning iron fertilisation for the mitigation of climate change
The imminence of global climate change is becoming politically accepted as its likely effect
on the world’s economy and environment. Despite this, there is still little suggestion that
political forces will engage the issue of fossil fuel consumption that lies at the heart of the
problem and it is therefore more and more likely that quick-fix, band-aid solutions will be
trialled. Already, entrepreneurial peoples have harnessed the desire of concerned individuals
and companies wishing to prove there environmental commitment, by developing tree
plantations throughout the world such as Future Forests which holds the registered trademark
for the term Carbon Neutral® (Future Forests Ltd., 2003). Such plantations are intended to
supply ‘carbon credits’ whereby the amount of carbon emitted is considered to be balanced by
the amount sequestered in the trees. There is still scientific uncertainty over whether this
provides a real and long-term sequestration option, and it also clear that the amount of tree
planting being carried out is more of a token effort than the actual amount of reforestation
required to combat the problem (Schimel et al., 2001).
The same approach is likely to happen with iron fertilisation. To date, there are several
patents already taken out by private individuals and companies on specific techniques for the
application of iron to the ocean, many of which are not much more complicated than spraying
acidified FeSO4.7H2O from a boat (Chisholm, 2000). One example is an enterprise known as
GreenSea Venture, Inc. which has recruited leading oceanographers and proposed an 8000
km2 demonstration experiment (Chisholm, 2000). Such proposals are attractive because they
harness the ‘carbon credit’ system of the post-Kyoto environmentalism (Jepma and van der
Gaast, 1998). It is intended that these private firms would charge the U.S government up to
$10 per tonne of carbon removed from the atmosphere via carbon drawdown through iron
fertilisation (McKie, 2003).
The scientific credentials of these proposals quote the results of the successful in-situ
experiments conducted so far. There has been little discussion, however of the effects that this
proposal will have on the ecological community composition of the areas where iron is
repeatedly supplied, and over the areas proposed. All the in-situ experiments have noted a
significant floristic shift towards a monoculture comprised of large phytoplankton species as a
result of iron fertilisation (Boyd et al., 2000, Coale et al. 1996, Coale et al., 1998). Such
![Page 68: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/68.jpg)
Modelling Ocean Fertilisation Patch Dynamics Discussion
60
changes are likely to have flow on effects to higher trophic levels, both those that have
already been documented, for examples changes in herbivory patterns, (Boyd et al., 2000,
Coale et al., 1996) and those that cannot foreseen.
Despite the associated uncertainties, iron fertilisation for mitigating global climate change is
still likely to go ahead at some stage. The results of the current study should not be viewed
only as the most economical way to facilitate the application of these fertilising projects, but
rather the best way to minimise the damage caused by the projects. If the same amount of
carbon can be sequestered by applying less iron over a shorter length of time and over a
smaller area, the ecological destabilising effects can be reduced. This is perhaps, a naïve view
considering from an economic perspective it is still more beneficial to apply more nutrients in
a more efficient manner. However, it does provide avenues for best practice and appropriate
regulation if commercial ocean fertilisation were to be accepted.
5.4 Recommendations
The approach taken in characterising the dynamics of export in the growth-diffusion-export
model in the current study has been primarily a qualitative one. The results presented are all
means of ten passes of the model under the same input parameters. Although some
quantitative analysis has been attempted by presenting the means of these ten passes, there is
often little evidence of patterns when parameters are varied. Considering the nature of the
export, modelled essentially by a stepped decrease in phytoplankton concentration, the export
flux often appears as instantaneous, where the duration of export is the smallest possible,
merely the time step assigned. Since the random seeding of the initial phytoplankton can
cause export events to vary, at least by a single time step, averaging over 10 passes may be
unrepresentative of the export flux and the duration of flux. Large peaks, for example of 100
mmol C m-2 day-1 that occur at the 4th time step in 1 pass and in the 5th time step in another
will be averaged to occur at a flux of 50 mmol C m-2 day-1 over both the 4th and 5th time step.
Thus the flux is reduced but the duration is increased. This effect will not be considerable but
may account for the unpredictable behaviour often seen in export time series.
Combining the passes of the runs could be done by other techniques, for example moving
averages, however, a more appropriate technique, given the random nature of the initial
phytoplankton distribution would be to identify important parameters and consider each of
these statistically, quantifying the mean, and variance. This approach would enable better
![Page 69: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/69.jpg)
Modelling Ocean Fertilisation Patch Dynamics Discussion
61
application of the results for those who wish to predict the behaviour of a phytoplankton patch
that has had iron applied to it.
It has been noted that the size of the subpatches of the fertilising patch are important in
governing the amount of export occurring. The approach taken here is by quantifying this
subpatch size by its length scale relative to the length of the entire patch. Since the export
generated is due to the relative areas occupied by fast growing phytoplankton (areas fertilised)
and limited growing phytoplankton (not fertilised) an area ratio approach may be another
approach which may better correlate the quantified fertilising regime with the amount of
export generated, and thus provide a more predictive method on what results might follow the
application of certain fertilising patches.
Although this study has been developed considering iron-depleted waters, the fertilising
nutrient applied during passes of the model could equally be any other limiting nutrient in a
different system with the same result. In reality only oceanic waters could be represented by
the length scales used here, or the use of length scale dependent diffusion, however nutrient
limitation is common in many inland waters, notably rivers and lakes. Modifications of this
model would enable it to be applied to such cases. The most obvious modification is that the
assumption of negligible vertical diffusion in and out of the mixed layer must be removed.
Adding the necessary third dimension would be greatly aided by incorporating the work of
O’Brien et al. (2003). Removing the assumption of the negligible vertical diffusion in and out
of the mixed layer, this model could be applied not only to inland waters where phytoplankton
dynamics are important, but also to larger length scales in the open ocean where vertical
diffusion becomes comparable to horizontal diffusion (Waite and Johnson, 2003).
Predicting the temporal behaviour is important in judging the most effective time to reapply
iron to the system. Power spectral density analysis is a powerful tool for identifying the
periods that are dominant in the time series behaviour of export, however, only one pass has
been conducted, using the basic parameters, that is sufficiently long to allow reliable analysis.
Future work would benefit from identifying the dominant periods that exist in the variations
of the fertilising concentration distributions.
As this study has been primarily a qualitative, considering the relative export capabilities of a
number of variations, there has been only limited discussion of real world comparisons. In
order to move beyond the qualitative approach it is necessary to calibrate the model to a
![Page 70: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/70.jpg)
Modelling Ocean Fertilisation Patch Dynamics Discussion
62
variety of conditions. The most significant parameter sets to investigate would be IronEx II
and SOIREE, to identify if the export generated by the model is of the same order as that
demonstrated in the in-situ experiments. Initial calibrations would then enable the
investigation of the reasons why the export was reduced so significantly in SOIREE in
comparison to IronEx II. Maximum growth rate and lateral diffusion are two parameters that
have been proposed thus far, for this discrepancy (Boyd, 2002). Calibration could also be
made to simulate the Aeolian deposition of iron to the HNLC areas, potentially investigating
the viability of Martin’s hypothesis using a growth-diffusion-export model.
Finally, it is recommended that other limiting factors be incorporated into the model to enable
a more realistic analysis of the phytoplankton and export dynamics at higher fertilising
concentrations.
![Page 71: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/71.jpg)
Modelling Ocean Fertilisation Patch Dynamics References
63
Chapter 6 Conclusion
If ocean fertilisation is to be successfully applied to mitigate climate change, the objective of
sequestering large amounts of carbon in the deep ocean will need to be done in a way that
makes the most efficient use of the nutrients applied. This study has used a growth-diffusion
and export model to interrogate a number of parameters which regulate the amount of export
that is generated from applying a nutrient patch; and how the export flux behaves over the
time of fertilising.
As maximum growth rate increases so does the amount exported. It is proposed that it does so
by allowing not only the centre of the patch, but areas towards the edge of the patch, to be in a
state where increases in concentration due to growth are greater than the losses due to
diffusion. However, in practice µmax cannot be manipulated to achieve greater export, it is a
property of the natural system to which it is applied. Understanding µmax does however enable
reasonable predictions of the timing of export events that are important to managing the state
of the enriched patch. The value of µmax is related to the timing of the first export event and is
also related to the timing of the complete depletion of nutrients if µmax and the fertilising
concentration are sufficiently small.
The parameter that can be most easily be manipulated to achieve the greatest export is the
fertilising concentration and how it is applied. Large concentrations of the fertilising nutrient
generate the greatest amount of export because it means that even with diffusive losses at the
edges of the patch, and multiple export events in the centre of the patch, sufficient nutrients
remain to encourage growth. If fertilising concentrations are sufficiently high, multiple export
peaks are induced and this means that the timing of some events such as the time to nutrient
depletion are no longer represented merely by µmax.
If ocean fertilisation was conducted, this study suggests that the most efficient use of iron
applied in generating carbon exported from the mixed layer will occur if the iron is applied at
high concentrations in localised areas rarely, rather than smaller concentrations spread over a
larger domain and applied more often. There will be a maximum concentration that it is
![Page 72: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/72.jpg)
Modelling Ocean Fertilisation Patch Dynamics References
64
feasible to apply, however, since other nutrients may become limiting. Characterising how
export will occur over time will depend both on µmax, the fertilising concentration, the period
between fertilising events and the length of the fertilising patch. The non-dimensional
parameter, Q needs to be used in addition to other parameters to properly describe the time
series behaviour of export so that adequate predictions can be used in management of ocean
fertilisation operations.
![Page 73: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/73.jpg)
Modelling Ocean Fertilisation Patch Dynamics References
65
References
Abraham, E., Law, C., Boyd, P., Lavender, S., Maldonado, M., Bowle, A., 2000. Importance of
stirring in the development of an iron-fertilized phytoplankton bloom. Nature 407: 727-730.
Bacastow, R.B., Keeling, C.D. and Whorf, T.P. 1985. Seasonal amplitude increase in atmospheric
CO2 concentration at Mauna Loa, Hawaii, 1959-1982. Journal of Geophysical Research 90: 10,529-
10,540.
Boyd, P., 2002. The role of iron in the biogeochemistry of the Southern Ocean and equatorial Pacific:
a comparison of in situ iron enrichments. Deep-sea research II – Topical studies in oceanography 49:
1803-1821.
Boyd, P., Watson, A., Law, C., Abraham, E., Trull, T., Murdoch, R., Bakker, D., Bowle, A.,
Buesseler, K., Chang, H., Charette, M., Croot, P., Downing, K., Frew, R., Gall, M., Hadfield, M., Hall,
J., Harvey, M., Jameson, G., LaRoche, J., Liddicoat, M., Ling, R., Maldonado, M., McKay, R.,
Nodder, S., Pickmere, S., Pridmore, R., Rintoul, S., Saft, K., Sutton, P., Strzepek, R., Tanneberger, S.,
Turner, S., Waite, A., Zeldis, J., 2000. A mesoscale phytoplankton bloom in the polar Southern Ocean
stimulated by iron fertilization. Nature 407: 695-702.
Boyd, P. and Law, C., 2001. The Southern Ocean Iron RElease Experiment (SOIREE) - introduction
and summary. Deep-sea research part II-Topical studies in oceanography 48 (11-12): 2425-2438.
Boyd, P., Jackson, G., Waite, A., 2002. Are mesoscale perturbation experiments in polar waters prone
to physical artifacts? Evidence from algal aggregation modelling studies. Geophysical Research
Letters 29 (11): 1-4.
Cavender-Bares, K., Mann, E., Chisholm, S., Ondrusek, M., Bidigare, R., 1999. Differential response
of equatorial Pacific phytoplankton to iron fertilisation. Limnology and Oceanography 44(2): 237-246.
Chisholm, S., 1995. The iron hypothesis – Basic research meets environmental policy. Reviews of
Geophysics 33: 1277-1286 Part 2 Suppl. S.
Chisholm, S., 2000. Stirring times in the Southern Ocean. Nature 407: 685-687.
Chisholm, S., Falkowski, P., Cullen, J., 2001. Dis-crediting ocean fertilization. Science 294: 309-310.
![Page 74: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/74.jpg)
Modelling Ocean Fertilisation Patch Dynamics References
66
Coale, K., Johnson, K., Fitzwater, S., Gordon, R., Tanner, S., Chavez, F., Ferioli, L., Sakamoto, C.,
Rogers, P., Millero, F., Steinberg, P., Nightingale, P., Cooper, D., Cochlan, W., Landry, M.,
Constantinou, J., Rollwagen, G., Trasvina, A., Kudela, R., 1996. Nature 383: 495-501.
Coale, K., Johnson, K., Fitzwater, S., Blain, S., Stanton, T., Coley, T., 1998. IronEx-I, an in situ iron-
enrichment experiment: Experimental design, implementation and results. Deep-sea research Part II-
Topical studies in oceanography 45: 919-945.
Crowley, T., 2000. Causes of climate change over the past 1000 years. Science 289: 270-277.
Cullen, J., 1995. Status of the iron hypothesis after the Open-Ocean Enrichment Experiment.
Limnology and Oceanography 40 (7): 1336-1343.
de Baar, H., de Jong, J., Nolting, R., Timmermans, K., van Leeuwe, M., Bathmann, U., van der Loeff,
Rutgers, M., Sildam, J., 1999. Low dissolved Fe and the absence of diatom blooms in remote Pacific
waters of the Southern Ocean. Marine Chemistry 66: 1-34.
Duce, T. and Tindale, N., 2001. Atmospheric transport of iron and its deposition in the ocean.
Limnology and Oceanography 36(8): 1715-1726.
Frost, B. Phytoplankton blooms on iron rations. Nature 383. 475- 476.
Future Forests Ltd., 2003. Future Forests for a CarbonNeutral world. [Online], Available from:
<http://www.futureforests.com> [1/11/2003]
Garret, C., 1983. On the initial streakiness of a dispersing tracer in two and three dimensional
turbulence. Dynamics of Atmospheric Ocean 7: 265-277.
Gordon, R., Johnson, K., Coale, K., 1998. The behaviour of iron and other trace elements during the
IronEx I and Plumex experiments in the equatorial Pacific. Deep-sea research II-Topical studies in
oceanography 45: 995-1041.
Gran, H., 1931. On the conditions for the production of phytoplankton in the sea. Rapports et process
verbaux des Reunions, Conseil International pour l’exploration de la Mer 75: 37-46.
Hart, T., 1934. On the phytoplankton of the Southwest Atlantic and the Bellingshausen Sea 1929-
1931. Discovery Reports 8: 1-268.
![Page 75: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/75.jpg)
Modelling Ocean Fertilisation Patch Dynamics References
67
Hutchins, D. and Bruland, K., 1998. Iron-limited diatom growth and Si:N uptake ratios in a coastal
upwelling regime. Nature 393: 561-564.
Jackson, G., 1990. A model of the formation of marine algal flocs by physical coagulation processes.
Deep-sea research 37(8): 1197-1211.
Jackson, G. and Lochmann, S., 1992. Effect of coagulation on nutrient and light limitation of an algal
bloom. Limnology and Oceanography 37 (1): 77-89.
Jepma, C. and van der Gaast, W., 1998. On the potential of flexible instruments under the Kyoto
protocol. International Journal of Environment and Pollution. 10(3-4): 476-484.
Kierstad, H. and Slobodkin, L., 1953. The size of water masses containing plankton blooms. Journal of
Marine Research 12:141-147.
Landry, M., Ondrusek, M., Tanner, S., Brown, S., Constantinou, J., Bidigare, R., Coale, K., Fitzwater,
S., 2000. Biological reponse to iron fertilisation in the eastern equatorial Pacific (IronEx II). I.
Microplankton community abundances and biomass. Marine Ecology Progress Series 201: 27-42.
Martin, J., 1990. Glacial-interglacial CO2 change: The iron hypothesis. Paleoceanography. 5 (1): 1-13.
Martin, J. and Chisholm, S., 1992. Design for a mesoscale iron enrichment experiment. Report of a
meeting held in Monterey, California, October 21 and 22, 1991. U.S JGOFS Planning Report, Vol. 15.
McCarthy, J., 1981. The kinetics of nutrient utilization, in Physiological Bases of Phytoplankton
Ecology, Platt, T., (ed.), pp. 211-233, Canadian Bulletin of Fisheries and Aquatic Science 210: 346p.
McKie, R., 2003. ‘Bid to reduce greenhouse gases ‘is folly’’. The Observer, UK. 12th January.
Morel, F., Hudson, R., Price, N., 1991. Limitation of productivity by trace metals in the sea.
Limnology and Oceanography. 36 (8): 1742-1755.
Muggli, D., Lecourt, M., Harrison, P., 1996. Effects of iron and nitrogen source on the sinking rate,
physiology and metal composition of an oceanic diatom from the subarctic Pacific. Marine Ecology
Progress Series 132: 215-227.
![Page 76: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/76.jpg)
Modelling Ocean Fertilisation Patch Dynamics References
68
O'Brien, K.R., Hamilton, D.P., Ivey, G.N., Waite, A.M., Visser, P.M., 2003. Simple mixing criteria for
the growth of negatively buoyant phytoplankton. Limnology and Oceanography 48(3): 1326-1337.
Okubo, A., 1971. Oceanic diffusion diagrams. Deep-sea research 18:789-802.
Platt, T., 1975. The physical environment and spatial structure of phytoplankton populations. Mem
Soc R Sci Liege 6e ser 7: 9-17.
Platt, T. and Denman, K., 1975. A general equation for the mesoscale distributor of phytoplankton in
the sea. Mem Soc R Sci Liege 6e ser 7: 31-42.
Sarmiento, J. and Bender, M., 1994. Carbon biogeochemistry and climate change. Photosynthesis
Research 39, 209-234.
Schimel DS, House JI, Hibbard KA, Bousquet P, Ciais P, Peylin P, Braswell BH, Apps MJ, Baker D,
Bondeau A, Canadell J, Churkina G, Cramer W, Denning AS, Field CB, Friedlingstein P, Goodale C,
Heimann M, Houghton RA, Melillo JM, Moore B, Murdiyarso D, Noble I, Pacala SW, Prentice IC,
Raupach MR, Rayner PJ, Scholes RJ, Steffen WL, Wirth C, 2001. Recent patterns and mechanisms of
carbon exchange by terrestrial ecosystems. Nature 414 (6860): 169-172.
Stanton, T., Law, C., Watson, A., 1998. Physical evolution of the IronEx I open ocean tracer patch.
Deep-sea research II-Topical studies in oceanography 45:947-975.
Trull, T., Rintoul, S., Hadfield, M., Abraham, E., 2001. Circulation and seasonal evolution of polar
waters south of Australia: Implication for iron fertilization of the Southern Ocean. Deep-sea research
II-Topical studies in oceanography 48 (11-12): 2439-2466.
Waite, A., Gallager, S., Dam, H., 1997. New measurements of phytoplankton aggregation in a
flocculator using videography and image analysis. Marine Ecology Progress Series 155: 77-88.
Waite, A. and Nodder, S., 2001. The effect of in situ iron addition on the sinking rates and export flux
of Southern Ocean diatoms. Deep-sea research II-Topical studies in oceanography 48 (11-12): 2635-
2654.
Waite, A. and Johnson, D., 2003. Critical space scales for aggregation-mediated carbon export from
ocean fertilization. Geophysical Research Letters 30(13): 1690-1694.
![Page 77: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/77.jpg)
Modelling Ocean Fertilisation Patch Dynamics References
69
Watson, A., Ledwell, J., Sutherland, S., 1991. The Santa Monica basin tracer experiment: Comparison
of release methods and performance of perfluorodecalin and sulphur hexafluoride. Journal of
Geophysical Research 96: 8719-8725.
Wigley, T. and Schimel, D., 2000. The Carbon Cycle. Cambridge University Press, Cambridge.
Wroblewski, J., O’Brien, J., Platt, T., 1975. On the physical and biological scales of phytoplankton
patchiness in the ocean. Mem Soc R Sci Liege 6e ser 7: 43-58.
Wroblewski, J. and O’Brien, J., 1976. A spatial model of phytoplankton patchiness. Marine Biology
35:161-175.
![Page 78: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/78.jpg)
Modelling Ocean Fertilisation Patch Dynamics Appendices
70
Appendix 1 “diffgrow2.m” script for growth-diffusion-export model
function [Cout,Eout,Nout,Et,tNdeplete,Cc,Cpua]=diffgrow2(C,Nf,Nb,K,G,T,dx,dt,steps,bound,movname,p) %diffgrow2 - diffgrow with nutrient limitation %usage: = %[Cout,Eout,Nout,Et]=diffgrow2(C,Nf,Nb,K,G[3],T,dx,dt,steps,bound,) % %solves the diffusion-growth-threshold equation: %dC/dt = K*del^2(C) + m*C - F(T,C) %over an arbitrary rectangular domain of the same size as input concentration matrix [C] with %grid size of [dx] (same for x and y) for [steps] timesteps of size [dt] % %[K] is the diffusion coefficient, and m is the growth rate - this is determined by the Monod equation: %m=G(1)*(N/G(2)+N); %where N is the total nutrient N=Nf+Nb , [G(1)] is the maximum growth rate %and [G(2)] is the half-saturation constant. %[G] contains the growth parameters and must be a three component vector % %Nutrients are put in as a background quantity [Nb] which remains constant %and a fertilized amount[Nf] which is used up as plankton grow and obeys: %d(Nf)/dt = K*del^2(Nf+Nb) - G(3)*m*C %where [G(3)] percentage nutrient usage per unit growth. % %[T] is the threshold criteria %so that: C=0 if C>=T % %[bound] defines the boundary condition (0=open 1=closed 2=periodic) %[movname] required for labelling movie %[p] for plotting and movie p=1. no plotting or movie p=0 % %[Cout] returns final concentration %[Eout] returns total removal due to threshold in a matrix same size as [Cout]; %[Nout] returns total remaining fertilizer %[Et] returns the timeseries of total export %[tNdeplete] returns time at which nutrient becomes depleted %[Cc] returns the timeseries of concentration over the entire test area %[Cpua] returns the time series of the mean concentration over the domain % %By default the plankton color range is set to [0 max(max(C),T)] and the nutrient color to max(max(Nf)); % %To view demo, type diffgrow('demo'); tNdeplete=0; %plotting switch: 1 for plotting, 0 for no plotting plotting=p; if nargin==1 if strcmp(C,'demo'); demo=1; else
![Page 79: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/79.jpg)
Modelling Ocean Fertilisation Patch Dynamics Appendices
71
demo=strcmp('Yes',questdlg('Run demo?','Demo')); end if (demo) C=rand(61);Nf=zeros(61);Nf(20:40,20:40)=0.9;Nb=0.01;K=0.1;G=[0.2 0.2 0.2];T=3.0;dx=1;dt=0.5;steps=100;bound=2; UIWAIT(msgbox({'Demo:';... 'Fertilization of a random plankton concentration with a square blob of fertilizer of concentration 0.9';' ';... 'Running demo simulation on a 61x61 grid with periodic boundaries';'using normalised parameters of:';... 'Diffusion coefficient K=0.1';'Growth parameters G = [0.2 0.2 0.2]';'Threshold parameter T=3.0;';... 'Input plankton concentration random values between 0 and 1'},'Diffgrow2 Demo','modal')); else return end elseif(nargin~=12) %fix up for different parameter numbers error('Diffgrow: Incorrect number of input parameters'); end [nx,ny]=size(C); nn=nx*ny; Cmax=max(max(max(C)),T); Nmax=max(max(Nf))+Nb; if plotting==1 Ff1=figure;set(Ff1,'NumberTitle','off','Name','Diffgrow2','Position',[50 120 780 500]); end %generate general coefficient matrix: K=K*dt/(dx*dx); m=G(1)*dt; AC=coeffmat(K,0,bound,nx,ny); AN=coeffmat(K,0,bound,nx,ny); %make plotting grid if plotting==1 [yy,xx]=meshgrid(1:dx:dx*ny,1:dx:dx*nx); end Nb=max(Nb,0.000001); Ct=reshape(C,nn,1); Nbmat=Nb*ones(nn,1); Nt=reshape(Nf,nn,1)+Nbmat; E=zeros(size(Ct)); if plotting==1 set(gcf,'DoubleBuffer','on') mov=avifile([ movname '.avi'], 'compression','indeo5', 'fps', 7); end for t=1:steps %calculate growth and m=G(1)*(Nt./(G(2)+Nt)); P=G(3).*Ct.*m; AC=spdiags(1+4*K-0.5*m*dt,0,AC);
![Page 80: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/80.jpg)
Modelling Ocean Fertilisation Patch Dynamics Appendices
72
DC=(1+0.5*dt*m).*Ct; DN=Nt-dt*P; %solve with Matlab Preconditioned Conjugate Gradient method [Ct,flag1] = pcg(AC,DC); [Nt,flag2] = pcg(AN,DN); if (flag1>0)|(flag2>0) warndlg('Solution failed - input parameters may be bad'); return end %----------- Cc(t)=sum(sum(Ct)); %threshold criteria - remove appropriate and store exported values fallout=(Ct>=T); E(fallout)=E(fallout)+Ct(fallout); Et(t)=sum(sum(Ct(fallout))); Ct(fallout)=0; %---------- Cpua(t)=mean(mean(Ct)); %maintain background level of nutrient Nt=max(Nt,Nbmat); if Nt<=Nbmat & tNdeplete==0 tNdeplete=t; end %plot results if plotting==1 figure(Ff1); set(gcf,'MenuBar', 'none') subplot('position',[0.05 0.3 0.4 0.6]) s=pcolor(xx,yy,reshape(Ct,nx,ny)); set(s,'FaceColor','Interp','LineStyle','none', 'Erasemode', 'normal'); axis equal; axis([1 dx*nx 1 dx*ny]);caxis([0 Cmax]);colormap(winter);%colorbar('vert'); title('Plankton concentration'); xlabel('Arbitrary units'); drawnow; subplot('position',[0.55 0.3 0.4 0.6]) s=pcolor(xx,yy,reshape(Nt,nx,ny)); set(s,'FaceColor','Interp','LineStyle','none','Erasemode', 'normal'); axis equal; axis([1 dx*nx 1 dx*ny]);caxis([0 Nmax]);colormap(winter);%colorbar('horiz'); title('Nutrient concentration'); xlabel('Arbitrary units'); subplot('position',[0.05 0.05 0.9 0.15]); plot([1:t],Et,'k-', 'Erasemode','none'); axis([1 steps 0 200]); %set(gca,'Color','none') title('Total export'); xlabel('Time (arbitrary units'); drawnow; F=getframe(gcf); mov=addframe(mov, F); end
![Page 81: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/81.jpg)
Modelling Ocean Fertilisation Patch Dynamics Appendices
73
end if plotting==1 mov=close(mov); end Eout=reshape(E,nx,ny); Cout=reshape(Ct,nx,ny); Nout=reshape(Nt,nx,ny); function A=coeffmat(K,m,bound,nx,ny) %function to construct the coefficient matrix nn=nx*ny; C1=(4*K-0.5*m+1); C2=-K; D=1+0.5*m; D1=C1*ones(nn,1); D2u=C2*ones(nn,1); D2l=D2u; D2l(nx:nx:nn)=0; D2u(1:nx:nn)=0; D3l=C2*ones(nn,1); D3u=D3l; %open boundaries if(bound==0) A=spdiags([D3l D2l D1 D2u D3u],[-nx -1 0 1 nx],nn,nn); %closed boundaries elseif(bound==1) D1([2:nx-1 nx+1:nx:nn-nx+1 nn-nx+2:nn-1])=C1-K; D1(nx+nx:nx:nn)=C1-K; D1([1 nx nn-nx+1 nn])=C1-2*K; A=spdiags([D3l D2l D1 D2u D3u],[-nx -1 0 1 nx],nn,nn); %periodic boundaries elseif(bound==2) D4u=zeros(nn,1); D4l=D4u; D4l(1:nx:nn)=C2; D4u(nx:nx:nn)=C2; D5l=C2*ones(nn,1); D5u=D5l; A=spdiags([D5l D3l D4l D2l D1 D2u D4u D3u D5u],[-nn+nx-1 -nx -nx+1 -1 0 1 nx-1 nx nn-nx+1],nn,nn); end
![Page 82: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/82.jpg)
Modelling Ocean Fertilisation Patch Dynamics Appendices
74
Appendix 2 “difffert.m” script for multiple fertilisation events
function [Earea,Etlong,Cclong,Cpualong,Nzero,dt,sumsteps]=difffert(fert,L,alph,g,Nn,C,p) % difffert - runs diffgrow specified number of times % usage: = % [Earea,Etlong,Cclong,Cpualong,Nzero,dt,sumsteps]=difffert(fert,L,alph,g,Nn,C,p) % % Applies nutrients in the concentration pattern given by [Nn], a 10 x 10 matrix over an initial % phytoplankton background given by [C] (50 x 50 matrix). Nutrient fertilisation is done initially % and then at intervals after this denoted by the elements of [fert]. The maximum growth rate is % given by [g] and the [alph] denotes the coefficient in the definition of length scale dependent % horizontal diffusion (see diffgrow2.m). p switches the plotting and movie creation to on (p=1) % or off (p=0). % % [Earea] gives the spatial distribution of total export % [Etlong] returns the time series of total export % [Cclong] returns the time series of total concentration % [Cpualong] returns the time series of concentration per unit area % [Nzero] returns a vector of the times at which nutrients became depleted to zero % [dt] returns the timestep used % [sumsteps] returns the total number of timesteps evaluated movname='pres_movie'; sumsteps=0; %defining length scale dependence K=alph*L^(4/3); dx=L/10; dt=min(0.5*dx*dx/K,0.2); %growth terms G(1)=g; %maximum growth rate G(2)=0.2; %1/2 saturation 3* Nb G(3)=0.2; %uptake ratio iron % fertilizer iron in uM/m3 Nnb=0.0; %concentration of fertilisation in non-targeted cells Nb=0; %background concentration in environment Nf=Nnb*ones(50); Nf(21:30,21:30)=Nn; %fertilised cells %plankton conc in mM C m-3 T=14; %threshold Etlong=[]; Cclong=[]; Cpualong=[]; Earea=zeros(50);
![Page 83: Modelling Patch Dynamics During Ocean Fertilisation](https://reader036.fdocuments.in/reader036/viewer/2022062507/62b0e0096d0ec51f7d052601/html5/thumbnails/83.jpg)
Modelling Ocean Fertilisation Patch Dynamics Appendices
75
for i=1:length(fert) steps=floor(fert(i)/dt); sumsteps=sumsteps+steps; %for plotting later [Cout,Eout,Nout,Et,tNdeplete,Cc,Cpua]=diffgrow2(C,Nf,Nb,K,G,T,dx,dt,steps,2,movname,p); Earea=Earea+Eout; Etlong(end+1:end+length(Et))=Et; Cclong(end+1:end+length(Cc))=Cc; Cpualong(end+1:end+length(Cpua))=Cpua; Nzero(i)=tNdeplete*dt; %refertilisation C=Cout; reNn=Nn; reNf=Nnb*ones(50); reNf(21:30,21:30)=reNn; Nf=Nout+reNf; end